I've got enough of a collection to have games I consider to be infamous (as opposed to Infamous, which would be a different title altogether).
My copies of Clash at Demonhead and Final Fantasy Tactics were purchased by my father as complete surprise gifts, both of which he knew I wanted badly but couldn't find/afford at the time. I had a complicated relationship with my father, where he was extremely abusive but would occasionally enter "Happy Drunk" phases of his life, and he would try to connect with me during those phases, but I would never know how to react. Both times I got the gift I was more confused than anything, rather than appreciative.
Clash at Demonhead he somehow remembered something like three years after he heard I wanted it. I didn't mention it again in the meantime at all; I was a very small kid, and I saw QVC (a shopping TV channel for you younger folks) showcase a video game, that being Clash at Demonhead. I only remember seeing the game on TV because it was the second time I had seen a game on TV, I think.
Final Fantasy Tactics he just saw in a store after I had spent *years* trying to find it. He was just coming out of an Angry Drunk phase and I remember being kinda scared on the phone as he read the back of the box to me on the phone, because, like...I thought I might get in trouble somehow since he bought the game? It was a Complicated Relationship.
Xenogears was the first game I bought over this weird thing called "the Internet." I hid it in shame, because I was very Catholic at the time and I thought my parents (who couldn't have cared less, really) would find I owned something kinda sacrilegious. The irony was, of course, that it is only symbolically sacrilegious, while Final Fantasy Tactics, which didn't spook me at all, actually was very anti-Catholic Church.
And my copy of Castlevania: Symphony of the Night I kept underneath my childhood Bible. I loved the game, and this time it wasn't a religious thing, as I wasn't Catholic anymore after I finally found a darn copy, but it had the weirdest history of falling from whatever shelf it was on. Just it, never anything else. So I weighed it down with a Bible. I jokingly said it was because the previous owner (it was used) had drawn a giant swastika on the back, and I needed to show The Lord I was alright with Him. You would have bought the copy of Symphony of the Night with a giant swastika too if you had found a copy in PS3 days, complete in case, for 10 bucks.
EDIT: I just realized that all the 'infamous' copies of games in my collection happen to be really, really beloved games. Clash at Demonhead is the best Metroidvania on the NES, Final Fantasy Tactics I loved to think about (but only learned to love to play after Alexander O. Smith did the PSP version's translation and Tzepish released his quality of life romhack), Xenogears might be the best JRPG of all time, and Castlevania: Symphony of the Night is my "go to" game when I'm feeling low and need to just concentrate on something other than Life.