4.1. X-Ray Binaries
X-ray binaries (XRBs) are the observational manifestations of stellar-mass black holes undergoing disklike accretion of matter captured from a companion star. The mean rate of mass supply is typically far below E, but the outer disks undergo occasional instabilities  that dump matter into the central regions at a rate that can approach or even exceed the Eddington limit. These outbursts can last months, and usually follow a specific sequence of thermal and spectral "states" that has never been adequately explained. A schematic representation of the sequence of states in the luminosity–spectral "hardness" plane is shown in Fig. 2 [32, 12], which is annotated to illustrate a possible model for the origin of the observed hysteresis.
Figure 2. X-ray binaries during outburst exhibit a temporal sequence of spectral states in the plane of X-ray hardness vs. luminosity. A similar sequence is seen in the plane of rms variability vs. luminosity, with variability and hardness strongly correlated. The sense of the cycle is counterclockwise, implying that a hard spectrum with increasing luminosity precedes a rapid transition to a soft spectrum at roughly constant luminosity, etc. The origin of the hysteresis exhibited by the cycle, in which the hard to soft transition occurs at a higher luminosity than soft to hard, is an unsolved problem. Cartoons illustrating different stages of the cycle show the geometric/thermal characteristics of each state (thin radiatively efficient disk vs. radiatively inefficient hot flow), which are widely agreed on, along with a more speculative proposal for the correlated behavior of the magnetic flux.  Figure is reproduced from Ref. 12.
Although the cyclic evolution of XRB outbursts is not well understood, there is reasonable confidence about the physical nature of each state. [33, 34] As the luminosity increases at the start of the outburst, the observed X-rays have a very "hard" (i.e., biased towards high energies), "nonthermal" spectrum that looks nothing like that expected from a thin (radiatively efficient), optically thick accretion disk. This is usually interpreted as evidence for a radiatively inefficient, two-temperature flow occupying the region around the black hole. As the accretion rate increases, this hot region shrinks and collapses to form a radiatively efficient disk. Both the pre- and post-collapse states are highly variable, and show evidence for a relativistic jet that is thought to indicate strong magnetic activity. Whether this activity is tied to the accumulation of magnetic flux is a subject of current speculation. [35, 12] A hard spectral "tail," combined with evidence of thermal disk emission, suggests that a highly magnetized corona surmounts the disk. Only at the highest luminosities, exceeding a few percent of the Eddington limit, does the magnetic activity calm down (often following a dramatic, final outburst) and the flow resemble a standard thin disk with a thermal spectrum peaking at moderate X-ray energies.
As the accretion rate declines, the flow does not retrace its previous evolution. Rather than developing signs of strong magnetic activity and evaporating to form a hot,two-temperature state, the disk remains thin and rather quiescent, decreasing in luminosity while remaining radiatively efficient. The transition back to a hot accretion state with strong magnetic activity does not occur until the luminosity has dropped to a fraction of the luminosity at which the forward transition took place. This hysteresis, illustrated in Fig. 2, also occurs in XRBs containing a neutron star rather than a black hole,  so it is apparently intrinsic to the accretion flow and not the properties of the central object.
4.2. Active Galactic Nuclei
An active galactic nucleus (AGN) is produced when a central supermassive (M > 106 M⊙) black holes accretes matter from its host galaxy. The outer boundary conditions for AGN accretion flows are much less understood (and probably much more diverse) than those for XRBs. Any standard thin disk model set up to supply the requisite mass for a luminous AGN is predicted to become self-gravitating beyond a fraction of a parsec, suggesting that star formation would interrupt or at least modify the flow. [37, 38] Attempts to resolve this problem — such as thermal regulation of the disk through stellar processes, magnetically supported disks, [39, 40] and stochastic injection of matter in random orbits  — show varying degrees of promise, but none has been strongly supported by observations.
The diversity of boundary conditions, as well as the cooler temperatures that typically prevail in AGN accretion flows, presumably lead to the highly diverse observational manifestations of these objects.  Many AGN are apparently surrounded by a geometrically thick torus of opaque, dusty gas that obscures a direct view of the inner accretion flow and reprocesses its emission to longer wavelengths. Unobscured AGN often exhibit an apparently thermal emission component peaking in the ultraviolet or soft X-rays, as predicted for thin disks with luminosities of roughly a few percent Eddington around black holes in this mass range. Such AGN also exhibit Doppler broadened emission lines that result from reprocessing of the thermal UV and X-rays, as well as corresponding absorption lines whose large blueshifts indicate that they are produced in a cool, fast wind.
Other AGN appear to be rather radiatively inefficient, with little evidence for thermal disk emission and luminosities well below the Eddington limit. Such AGN are primarily detected through broadband emission from powerful, relativistic jets.  These are especially well-studied in the radio (mainly because radio telescopes are especially sensitive compared to telescopes operating in other bands), but also emit strongly in all other bands ranging up to X-rays and gamma-rays. The relativistic nature of these jets is established through clear signatures of Doppler beaming (e.g., one-sided structure on the sky and rapid variability) and apparent superluminal motion (an illusion due to light travel-time effects) in jets pointing close to the line of sight.
AGN exhibit variability but, unlike XRBs, seldom change their accretion state dramatically. It could be that the manner in which gas is supplied determines the long-term mode of accretion, e.g., disklike and radiatively efficient when interstellar clouds accumulate and settle into a disk, hot (two-temperature) and perhaps starlike when matter is supplied from a hot halo of gas surrounding the nucleus of the galaxy or from stellar winds, as is believed to be the case in the Galactic Center.  The fact that radiatively inefficient flows with powerful jets occur much more frequently in elliptical galaxies (which have predominantly hot interstellar matter) than in spirals (which have an abundant cool gas) seems to support this environmental interpretation. These differences in accretion mode could also affect the accumulation of magnetic flux, thus governing jet production indirectly.  But one cannot rule out the possibility that AGN undergo state transitions analogous to XRBs, as their vastly larger spatial scales suggest that such transitions would take place too slowly to detect.
4.3. Tidal Disruption Events
Tidal disruption events (TDEs) are transient episodes of accretion triggered when a star is partially or completely disrupted by the tidal gravitational field of a supermassive black hole. [46, 47] In a typical event about half the mass escapes, but the other half falls back gradually over time, with a rate scaling roughly as t-5/3 at late times. 
Only black holes less massive than about 108 M⊙ are capable of disrupting main-sequence stars; more massive holes would swallow such stars whole. A solar-type star would have to venture to within about 30 Schwarzschild radii of the Galactic Center's black hole (which has a mass of several million M⊙) in order to be disrupted, implying that the mean specific angular momentum of the debris would only be a few times GM / c. This means that fallback cannot lead initially to a disklike accretion flow that extends to large radii, because there is not enough angular momentum compared to the Keplerian value. At later times, however, as matter is swallowed leaving behind most of its initial angular momentum, the mean angular momentum per unit mass of the remaining material increases and the flow can evolve toward a more disklike state.
Despite the relatively small cross-section for tidal disruption, TDEs are expected to be fairly common in nearby galaxies, occurring once every 104 years or so in a galaxy like the Milky Way. About two dozen candidate TDEs have been identified through the soft X-ray thermal spectra predicted to characterize their accretion disks and their characteristic light curves, which peak days to months after the disruption (depending mainly on the mass of the black hole) and then decline roughly according to t-5/3. [49, 50]
Disruptions of solar-type stars by black holes in the mass range 105 - 106 M⊙ are predicted to lead to fallback rates large enough to produce super-Eddington luminosities for ∼ 1-3 yr, if the debris were accreted in real time. [51, 52] Two TDE candidates have been discovered which appear to exceed the Eddington limit by about two orders of magnitude (for the estimated SMBH mass of ∼ 106 M⊙), even after correcting for beaming. [53, 54, 55, 56] Their observed decay rates suggest that the luminosity tracks the fallback rate, and the presence of a radio afterglow in both cases suggests the production of a relativistic jet.
If these events represented disklike accretion, one would expect self-regulation of the mass flux reaching the black hole to a value that did not exceed E by a large factor. [57, 58] But super-Eddington TDE accretion flows are probably starlike, given the low specific angular momentum of the accreting matter and the fact that it is probably pushed out to rather large distances by the pressure of trapped radiation. In Sec. 3.2 we argued that radiatively inefficient, starlike flows are unable to regulate the rate at which matter reaches the black hole, but were unable to decide the outcome of the energy crisis that likely ensues. Observations of super-Eddington X-ray luminosities and jets from TDEs suggest that, at least in these systems, the excess energy finds a relatively stable escape route through the rotational axis. 
4.4. Gamma-Ray Bursts
The close coincidence between long-duration gamma-ray bursts (GRBs) and core collapse supernovae supports the collapsar model, in which the burst results from the formation and rapid growth of a black hole or neutron star at the center of a massive stellar envelope.  The long duration of such bursts (minutes or more) implies sustained accretion at an extremely high rate,  while the large total energies involved favor a black hole engine, at least for the most luminous bursts. The inferred accretion rates initially can be as large as a tenth of a solar mass per second. . While enormously super-Eddington (by up to 14 orders of magnitude), the initial episode of accretion is not necessarily radiatively inefficient, because neutrinos can carry away most of the liberated energy; however, as the accretion rate declines and neutrino losses become insignificant, the flow must revert to an extremely radiatively inefficient state.
Our ignorance about the angular momentum distribution inside the stellar progenitor make it difficult to determine whether the late-time accretion flow during a long-duration burst is disklike or starlike. The weight and optical depth of the overlying envelope certainly prevent the Eddington limit from being a factor in regulating the rate at which mass reaches the black hole — radiation is too thoroughly trapped. However, we would expect a disklike flow to adjust so that some outward advection or circulation of energy suppresses the accretion by about a factor (cs / c)2 below the Bondi value, where the latter is calculated using the self-consistent value of the density and sound speed cs at the "Bondi radius" GM / cs2 inside the stellar envelope. But the accretion rates needed to explain the prompt emission from long-duration bursts far exceed this, suggesting that no such regulation occurs. A starlike accretion flow, producing orders of magnitude more energy than can be wicked away by the accretion flow, would suffer a similar energy crisis to that in the jetted TDEs. . Like the TDEs, GRBs are evidently able to dispose of the excess energy through powerful jets punching through a quasi-spherical envelope.
These jets are remarkable for their enormous bulk Lorentz factors (Γ ∼ 100-1000) which are inferred from variability considerations and the requirement that the gamma-rays be able to escape. These Lorentz factors are 1–2 orders of magnitude higher than those found in other jets produced by black hole accretion, such as the jets from AGN and XRBs. I will comment on the possible significance of this below.