What is the Poincare recurrence of the OBSERVABLE universe? I have heard about a number called the Poincare recurrence time on Numberphile. It did not seem that exact in the way that it calculated this number, which was a power tower of 10s.
What is the Poincare recurrence of the observable universe, which is finite? (And assuming this universe is a Hamiltonian finite system.)
 A: The problem is that this is not a closed system. The observable universe is growing as time goes by and more and more photons arrive from remote parts of the past. So in a sense it doesn't have a recurrence time. Also, since the universe is expanding it is not clear phase space volume is conserved, so the theorem may not apply.
The recurrence time for a finite region of radius R and total mass M is bounded by the Bekenstein bound $I=k R M$ which tells us the number of bits needed to describe it. $\tau_1 \sim 2^I=2^{k MR}$ is the number of states the region could be in, and if we assume it is moving between distinguishable states this gives a timescale until there is some repeat - there has to be a return to one earlier state since there are no more untested states. If we exponentiate it again to $\tau_2 \sim 2^{2^{kMR}}$ we get a rough estimate of how long it takes to run through all the different evolutions randomly and end up at a particular starting point.
The time unit, whether Planck seconds or kalpas, does not matter much since the number of bits in the exponent is 10 to the power of something for any macroscopic system; double exponentials dominate over whatever time measure you use. This is also why it doesn't matter if the base is 10, e or 2 ("All e's become tens" as the video says).
This is much less than in Page's paper if you just use the observable universe, but in practice the magnitude does not matter. These kinds of magnitudes are interesting only when comparing to things of a far different magnitude. The details do not matter as much as the principles.
The longest timescales I have seen in physics papers that are not doubly exponential are some estimates of vacuum decay timescales running up to $10^{1000}$ years; this is much much smaller than $\tau_1$, which is much much much smaller than $\tau_2$.
