Well, this is by no means settled, but that's ok (meaning any answer is likely going to be seen as controversial until further observation provides more data).

John Rennie's answer doesn't appear to address whether or not singularities actually form in the first place; he does suggests that few skeptics would argue this, but in fact a few have (David Hilbert, for example, argued infinities cannot physically exist in nature, but merely as metaphysical constructs, and so would likely argue against a singularity with infinite space/time curvature.  Likewise Pawel Mazur and Emil Mottola in "Gravitational Condensate Stars: An Alternative to Black Holes" have assumed the same limitation as Hilbert and explored the theoretical consequence and come up with the 'Gravastar', a theoretical construct with properties similar to black-holes, and mathematically no less real (given that black-holes are only ever observed indirectly by the effect they have on surrounding space ).  Notwithstanding John Rennie's suggestion about singularities, he's otherwise made it pretty clear, if singularities do exist, they cannot do so without an event horizon.  That still doesn't address event horizons without singularities though.

Consider, as your falling clock falls toward an event horizon it gains speed, and loses thermal energy (else poses problems for the black-hole due to the 2nd law of thermodynamics).  As it falls more quickly, because its time slows, the rate at which it loses energy will appear to decrease.  It will appear to go the speed of light at the event-horizon but never appear to cross over because of its stopped time.  It should also not have any energy.  Because black-holes are not thought to break the 2nd law of thermodynamics it is generally believed necessary that all energy is radiated away through thermal or Hawking radiation before the event Horizon is reached, but for the sake of your question lets suppose this doesn't matter.  The point is that once it reaches the event-horizon the rate of energy loss must appear to be zero for, as you point out, this clock's time period compared to the remote clock observing, will be infinitely long.

Matter whose rate of energy absorption/energy emission is said to be effectively zero is recognized to be Bose-Einstein condensate - which raises a question about the nature of the event-horizon.  Paul R. Anderson, Roberto Balbinot, Alessandro Fabbri, Renaud Parentani and others have shown (Hawking radiation correlations in Bose Einstein condensates using quantum field theory in curved space) that a Bose-Einstein event horizon will produce Hawking radiation, and this has been observed (Hawking radiation in a two-component Bose-Einstein condensate, P.-É. Larré, N. Pavloff)

Although some have argued that 'infinite time' is an illusion caused by the falling clock red-shifting out of communication with the rest of the universe, it doesn't need to be mere illusion, but may actually be (well, as close as we can get at least) if all energy is radiated away prior to reaching the event-horizon causing the formation of Bose-Einstein condensate whose surface is the event-horizon itself.  This means that as the clock's period appears to slow it does so because all it's energy is being lost which suggests that not only is space time, but also that energy is as well - not a surprising result given super-string theory which posits matter and energy are one and the same.

By that interpretation, the observer clock is viewing what amounts to a massively slowed-down recording of the moments right up to the falling clock leaving the visible universe where the rate of emission of energy is directly related to the energy still contained in the clock as it falls.

This means then, with respect to your original question that; you can have event-horizons without singularities, albeit Bose-Einstein condensate ones known as 'Gravastars'; matter cannot actually fall through an event horizon, since the EH effectively represents a zero-energy boundary; and that infinite space time curvature and singularities need not exist for there to be event-horizons (Hilbert would be so pleased).

It also means that the falling clock is still available to interact with an observer clock a billion years from now, that it is not truly "in the black" but to do so would require it to be able to absorb energy.  The property of Bose-Einstein condensate with respect to Hawking radiation has been studied, both near and on the event-horizon itself, and although it is not impossible for this to happen, it is not very likely.