Identity Theft
Okay, this answer is pretty crazy, but hear me out. Our superhuman, perfect memory robot should take advantage of someone who already has a lot of wealth. Our robot just has to kidnap/kill (murder?) a current billionare. Preferably young, as that gives him the most amount of "years" to be that person.
In terms of actually stealing the identity, given that our robot has great memory and 100% of history, I assume he knows how to impersonate people well and make disguises (given that we don't know the "biological" make-up of our robot, I wont assume much more). As long as there isn't some biological password (and depends. Our robot could "borrow" the unfortunate's victim's eye or hand or etc. etc.) to this person's wealth, our robot will eventually figure out how to bypass the various security measures protecting the wealth.
To hide the fact that he isn't actually Mr. (or Ms.) Net-Worth-Billions, it would be best to choose someone who isn't a large personality with the media and has relatively no obligations/connections. No living nuclear family would be ideal. And if the robot wants, he could make this figure drop off the face of the earth (within a few years, with our robot's capabilities), and as the years go by change identities to legitimate people after his fake identity "dies", willing his fortune to his new self, of course. A quick google search provides a good list here. (though I kept my research to a minimum. I feel as if searching "single young billionaires" is a little weird...)
To further increase his fortune, do what the other answers have suggested: invest in stock, gamble, etc.
NOTE: This idea came to my head before reading James's answer, but it follows the idea of using someone who already has a great deal of wealth.