Well then, I can happily inform you that depending on just how physics works out, we might just get a solution to that whole "heat death of the universe" problem. Too early to tell, precisely, but I've heard some neat things about new child-universes forming inside black holes such that they torque space itself and generate non-conserved new mass-energy.
Any value of forever large enough that we need to find someplace to put our excess memories outside our original brain is still getting into "forever" territory from the perspective of our current lives.
People already do this. The traditional forms of memory storage are known as "diaries" and "photo albums". It is universally acknowledged that without their aid, your brain will overload and lose the information you wanted to preserve.
But despite all this, no one thinks of human life as lasting "forever", or anywhere close to it.
The irony is the only things I can't forget are the ones written down or photographed. I never need to look at them, but if they weren't written down it would have vanished.
It's like I have a form of telepathic cloud storage. So long as the physical information exists, I can always remember it and will never look. So some is potentially in a Schrödinger's cat state as I never know if my wife actually threw the note out or not as I'll never need to go look for it.
Second law of thermodynamics. Granted this is a bit pedantic since we are nowhere close to that as a limiting factor.