The first games that I ever rented were from my local Blockbuster (that's now an O'Reilly's Auto Parts store), a bunch of special SNES games: Super Star Wars, Boogerman, Earthworm Jim, and Rocko's Modern Life. Not sure why I picked those titles, but whevener my cousins would visit me and stay at my house, my Mom would take me over there to rent a game or two for the weekend. Didn't really have any "mom and pop" type video rental places, just a choice between Blockbuster and Hollywood video. My best friend was way more into renting games then I was, and had a few more consoles than I did. I was pretty much going to his house every weekend, so whatever he was renting from Blockbuster that weekend we were definitely playing all night.
Renting continued into the N64 era, where I rented tons of stuff, but most notably Hey You, Pikachu! and Pokemon Snap (which I ended up buying), and later PS2/Xbox era. My friend and I tried out Game Crazy at H.V. briefly, but pretty much stuck with the games club at Blockbuster because of the 1 or 2 game at a time deal. So legit.
After both closed (the Blockbuster first, followed by the family video, which is now a Rite Aid or something, that still has the Hollywood Hills and Spotlight trim on the side of the building), I did some brief game rentals from Family Video for the Xbox 360. But, between the terrible categorization in the store, the lack of selection, and the bad condition of the games there, it was just not worth it. But man, I miss those Blockbuster days sometimes, of how simple it was to just grab an N64 or SNES cart, pop it in, and have a weekend disappear.