The first problem is that, given that average inflation rates have exceeded 2% per year, any investment at 2% annually is going to be a terrible investment in the future - you're actually LOSING money every year on average rather than gaining it, in terms of actual value.
However, let us assume that for whatever reason you were actually getting 2% growth on top of inflation per year (an unlikely scenario for an unguided account, but bear with me a moment). The second problem is that the result is, obviously, irrational. There are not even a googol particles in the Universe; how could you have a googol dollars and have that be a reasonable result? There isn't anything to purchase with a googol dollars. Ergo, we must assume our assumption is flawed.
The flaw in this assumption is, of course, the lack of understanding of exponential growth; all exponential growth is self-limiting in nature. In reality you run into real constraints to how much growth you can have, how much of the economy can be in your fund, ect. As someone else pointed out, you can't assume that you will have indefinite growth; it is confined by the size of the economy (which will never reach a googol dollars in current day money), by the likelihood of you actually getting said returns (which is 0), whether anyone would recognize a currency when too much of the world economy was bound up into it, ect.
The truth is it is just a terrible argument to begin with. Anyone who promises you 2% growth for even a thousand years is a dirty liar. The rate seems reasonable but in actuality it is anything but. Do you think that the world economy is going to increase by 2% a year for the next 1000 years? I don't, at least not in real, inflation-adjusted dollars. The total amount of energy we can possibly use on the planet alone would constrain such economic growth.
Robin Hanson wrote, five years ago:
In the comments some people gave counterarguments. For those in a rush, the best ones are Toby Ord's. But I didn't bite any of the counterarguments to the extent that it would be necessary to counter the 10^100. I have some trouble conceiving of what would beat a consistent argument a googol fold.
Things that changed my behavior significantly over the last few years have not been many, but I think I'm facing one of them. Understanding biological immortality was one, it meant 150 000 non-deaths per day. Understanding the posthuman potential was another. Then came the 10^52 potential lives lost in case of X-risk, or if you are conservative and think only biological stuff can have moral lives on it, 10^31. You can argue about which movie you'll watch, which teacher would be best to have, who should you marry. But (if consequentialist) you can't argue your way out of 10^31 or 10^52. You won't find a counteracting force that exactly matches, or really reduces the value of future stuff by
3 000 000 634 803 867 000 000 000 000 000 000 777 000 000 000 999 fold
Which is way less than 10^52
You may find a fundamental and qualitative counterargument "actually I'd rather future people didn't exist", but you won't find a quantitative one. Thus I spend a lot of time on X-risk related things.
Back to Robin's argument: so unless someone gives me a good argument against investing some money in the far future (and discovering some vague techniques of how to do it that will make it at least one in a millionth possibility) I'll set aside a block of money X, a block of time Y, and will invest in future people 12 thousand years from now. If you don't think you can beat 10^100, join me.
And if you are not in a rush, read this also, for a bright reflection on similar issues.