One idea that was interesting (and I have thought myself), was whether some future super-civilization may be able to actually tinker with the laws of phyics, or otherwise influence events in a manner to cause the universe to collapse (if that is what they desired).
Hal Finney wrote:
Lee Corbin points to Tipler's March 2005 paper "The Structure of the World From Pure Numbers": http://www.iop.org/EJ/abstract/0034-4885/68/4/R04
I tried to read this paper, but it was 60 pages long and extremely technical, mostly over my head. The gist of it was an updating of Tipler's Omega Point theory, advanced in his book, The Physics of Immortality. Basically the OP theory predicts, based on the assumption that the laws of physics we know today are roughly correct, that the universe must re-collapse in a special way that can't really happen naturally, hence Tipler deduces that intelligent life will survive through and guide the ultimate collapse, during which time the information content of the universe will go to infinity.
The new paper proposes an updated cosmological model that includes a number of new ideas. One is that the fundamental laws of physics for the universe are infinitely complex. This is where his title comes from; he assumes that the universe is based on the mathematics of the continuum, i.e. the real numbers. In fact Tipler argues that the universe must have infinitely complex laws, basing this surprising conclusion on the Lowenheim-Skolem paradox, which says that any set of finite axioms can be fit to a mathematical object that is only countable in size. Hence technically we can't really describe the real numbers without an infinite number of axioms, and therefore if the universe is truly based on the reals, it must have laws of infinite complexity. (Otherwise the laws would equally well describe a universe based only on the integers.)
Another idea Tipler proposes is that under the MWI, different universes in the multiverse will expand to different maximum sizes R before re-collapsing. The probability measure however works out to be higher with larger R, hence for any finite R the probability is 1 (i.e. certain) that our universe will be bigger than that. This is his solution to why the universe appears to be flat - it's finite in size but very very big.
Although Tipler wants the laws to be infinitely complex, the physical information content of the universe should be zero, he argues, at the time of the Big Bang (this is due to the Beckenstein Bound). That means among other things there are no particles back then, and so he proposes a special field called an SU(2) gauge field which creates particles as the universe expands. He is able to sort of show that it would preferentially create matter instead of antimatter, and also that this field would be responsible for the cosmological constant which is being observed, aka negative energy.
In order for the universe to re-collapse as Tipler insists it must, due to his Omega Point theory, the CC must reverse sign eventually. Tipler suggests that this will happen because life will choose to do so, and that somehow people will find a way to reverse the particle-creation effect, catalyzing the destruction of particles in such a way as to reverse the CC and cause the universe to begin to re-collapse.
Yes, he's definitely full of wild ideas here. Another idea is that particle masses should not have specific, arbitrary values as most physicists believe, but rather they should take on a full range of values, from 0 to positive infinity, over the history of the universe. There is some slight observational evidence for a time-based change in the fine structure constant alpha, and Tipler points to that to buttress his theory - however the actual measured value is inconsistent with other aspects, so he has to assume that the measurements are mistaken!
Another testable idea is that the cosmic microwave background radiation is not the cooled-down EM radiation from the big bang, but instead is the remnants of that SU(2) field which was responsible for particle creation. He shows that such a field would look superficially like cooled down photons, but it really is not. In particular, the photons in this special field would only interact with left handed electrons, not right handed ones. This would cause the photons to have less interaction with matter in a way which should be measurable. He uses this to solve the current puzzle of high energy cosmic rays: such rays should not exist due to interaction with microwave background photons. Tipler's alternative does not interact so well and so it would at least help to explain the problem.
Overall it is quite a mixed bag of exotic ideas that I don't think physicists are going to find very convincing. The idea of infinitely complex natural laws is going to be particularly off-putting, I would imagine. However the idea that the cosmic microwave background interacts differently with matter than ordinary photons is an interesting one and might be worth investigating. It doesn't have that much connection to the rest of his theory, though.