Alright, this is kind of a Special Interest, so here's your relevant thought dump.

First up, the image is kind of misleading, in the sense that you can always tack on extra orders of magnitude. You could tack on another thousand orders of magnitude and make it look even longer, or just go "this is 900 OOM's of literally nothing happening, lets clip that off and focus on the interesting part"

Assuming proton decay is a thing (that free protons decay with a ridiculously long half-life)....

ok, I was planning on going "as a ludicrous upper bound, here's the number", but, uh, the completely ludicrous upper bound wound up being a WHOLE LOT longer than I thought. I... I didn't even think it was possible to stall till the evaporation of even a small black hole. But this calculation indicates that if you're aiming solely at living ludicrously long, you can stall about a googol years, enough for even the largest black holes to evaporate, and to get to the end of the black hole era. I'm gonna need to rethink some stuff.

EDIT: rethought some stuff, realized it doesn't change my conclusions from when I last looked into this. The fundamental problem is that, for any remotely realistic numbers, if you're trying to catch the final evaporation of a black hole to harvest its mass-energy, you'll blow a lot more than the amount of mass-energy that you could gain, in order to wait that long.

Final conclusion: If proton decay is a thing, it's definitely not worth waiting to the end of a black hole, you'll want to have things wrapped up far earlier. If proton decay isn't a thing, you'll want to wait till the black hole evaporates to catch that final party and last ${10}^{19}$ kg of mass-energy. If proton decay is a thing and you're willing to blow completely ridiculous cosmic amounts of resources on it, you can last till the late parts of the black hole era.

The rough rationale is as follows. Start with 10x the mass of the largest black holes in the universe, around ${10}^{12}$ solar masses stockpiled. If they're spinning fast enough, you can extract energy from them, assume you can extract all of it (it's over 10 percent, so let's round it up to 100 percent). Assume that proton decay is ${10}^{40}$ years (a high estimate), and that we use the energy at 100 percent efficiency to make matter (also high estimate), you can take out one proton, wait for around a proton decay time, take out the next proton, and so on. Then you can take out around ${10}^{69}$ protons, and each one lasts you around ${10}^{40}$ years, getting around ${10}^{109}$ years (high uncertainty). And, coincidentally, natural Hawing radiation finishes off the black hole of that size in ${10}^{103}$ years, leaving a small margin left over for silly considerations like "maybe the intelligence needs more than one proton to physically implement".

So, not remotely practical, but maybe something like ${10}^{80}$ years would actually be doable? That extra 29 OOM's of wiggle room patches over a lot of sins in this calculation.

But, in terms of what would actually be practical for the far future of humanity, it'd be the strat of "dump as much mass into a fast-spinning black hole as possible. Like, eat the entire Laniakea supercluster complex. Wait a trillion years for the cosmic microwave background radiation to cool to its floor temperature. You'd be in the late Stelliferous era at this point, with a few red dwarfs around, if you didn't dump all the stars in the mega-black-hole already. Set up some infrastructure around the mega-hole, and use the Blandford-Znajek mechanism to convert the mega-hole spin into electrical power. You should be able to get about a gigawatt of power for the next ${10}^{45}$ years to run a whole lotta computation and a little bit of maintanence, and if proton decay is messing with things, chop however many OOM's you need off the time and add those OOM's to the power output. Party for a trillion trillion trillion eons with your super-optimized low-temperature computing infrastructure"

That's an excellent question, pondered by the brightest minds. The great Freeman Dyson proposed a solution dubbed eternal intelligence (Dyson 1979, Reviews of Modern Physics, Volume 51, Issue 3, July 1979, pp.447-460). Basically, some finite amount of matter=energy is stored. As the universe cools over time, energy costs per computation decrease (logarithmically, but forever). After each cooling time period, one can use some fraction of the remaining energy, which will thus never go to zero, leading to eternal consciousness. 

It was later understood that the expansion of the universe is accelerating. If that holds, the concept breaks down, as Dyson admitted. In the far future, any two observers will be separated, making the remaining subjects very lonely. 

I tried to answer the question here: How to Survive the End of the Universe 

I'm not sure that you can get to 10^60 years even if protons don't decay. If they do, then you might not have very much more time than whatever the decay timescale is. It could be that free protons decay while bound ones don't, in which case you might just avoid having any hydrogen or other light elements around.

Assuming you have a galactic mass available to do with as you please and can extract essentially all of its mass-energy, then over 10^60 years you would have a few picowatts available to do all the maintenance that might be required for whatever infrastructure you have. That's a pretty miserly budget for managing structures at a scale that can convert galactic mass to usable energy! It would have to cover all sensors as well as computation and also any work that might need doing from time to time. I would be very surprised if maintenance costs didn't scale along with mass, just because the number of particles enormously expands the state space.

Over these sorts of timescales even the most rigidly bound materials diffuse like a gas due to various quantum processes, but maybe there is some clever way to avoid that and still have them able to perform the desired functions when required.

Anything over these sorts of timescales are going to be very speculative, so this is probably about the best "answer" I can give. In reality, we don't know what happens to matter over 10^30 year timescales even if the cosmological predictions were spot-on.

Assuming protons don't decay and that there's no big rip, I feel like you can do obnoxiously large numbers.  Build a clock out of superconductors that consumes 0 power until the top bit flips over (since incrementing a counter is reversible, this should be possible).  Then, when your "alarm" goes off, wake up the sentient being and let it have its last thought.  The limit is now on the number of bits B in your clock.  Assume this is somewhere between 10**67 (number of atoms in the galaxy) and 10**80 (number of atoms in the visible universe).  Your wake-up time is now 2**(10**B).

If there is a way for data structures to survive forever it would be something we couldn't imagine, like three leptons orbiting each other storing data in their precise separation distances, where it would take a godzillion eons to generate a single pixel in an ancient cat picture.