Brace yourself for this: the basic elements of this view, motivated by the second law of thermodynamics, include slowly decaying masses over cosmic time (so we cannot compute them), a Dark Energy and genuine information loss into black holes. Penrose gave a wonderful explanation of what entropy is, using the interaction of the sun and earth as an example of how important entropy is for understanding the cosmos. He explained how the CMB was black body radiation and that this meant a high entropy state, whereas the Big Bang should be a low entropy state. The resolution is to realise that gravity behaves very differently to other matter, and that gravity is also needed to describe the initial singularity. That is, entropy increases with gravitational clumping and so a smooth initial state can correspond to a low entropy state. Finally, the new ingredient is the idea that time loses its meaning in the early universe. Treated as a conformal boundary, matter should be considered massless and since such boundaries may be attached to earlier eons (he assumes a cosmic cyclic time) the singularity effectively disappears.
The mathematical argument hinges on the concept of conformal boundary, so Penrose went to the trouble of explaining Weyl curvature by showing images of distorted galaxies in gravitational lensing rings. The extra scalar factor needed to describe clocks in the massive regime arises as structure internal to all the light cones, namely a sequence of internal hyperboloids for ticks of a clock. In other words, clocks only make sense when there is mass, as indicated by frequency = m.constant (assuming hbar and c constant). It is necessary here for the late universe to also lose its concept of time, as everything forms black holes which only evaporate away incredibly slowly. Penrose's solution was to say, well, there will be no humans, just a bunch of photons, which don't see time anyway.
Of course, this idea is full of holes. The first thing one is tempted to do is to pile all the eons on top of each other and try to find some multiverse picture that still respects the second law in this elegant way. Perhaps this cosmology is useful in that sense. Penrose also predicted that conformal structure would be detected in the WMAP data, and perhaps this carries over to a more quantum viewpoint also. He did mention that a serious problem might arise from issues in the foundation of quantum mechanics. I think the biggest conceptual issue is the lack of a concept of 'NOW'. In a cosmology with time asymmetry, any observer must have a means of distinguishing the past from the future, and this suggests a time mark with the meaning of now, but in Penrose's picture there is nothing special about any given era within the cosmic epoch, and there is no quantitative means of measuring the complexity of interesting physical systems, such as life.