If I had to pick a game, it would be Twinkle Star Sprites in MAME. I played around with a handful of titles before that, but that was the first one to really capture my attention. Metal Slug X and Windjammers are other possible choices, but I recall it being TSS.
To be honest, I think the obsession started before playing any games, just based on what the hardware itself is and the history behind it. My dad bought me an SNES when I was three, because he wanted one and figured my mom wouldn't complain if he bought it for the kids; to his credit, he really did give it to us, I just assume he played it when we went to bed, because, you know, little kids. Anyway, as I got older, just read around a lot on video game stuff, and eventually heard of this system called the Neo Geo. Which was, somehow, an arcade machine and a home system - at the same time.
"Wait, what the fuck? That doesn't make any sense."
Kept collecting, and just never saw one at thrift stores flea markets, or anywhere; honestly, I've never even seen any AES (or CD stuff, for that matter) at a game store, even. First time I got to play on legitimate hardware was when my dad bought an MVS cab about a decade back with a two-slot board containing Bust-A-Move and Street Slam, which he proceeded to gut and attempt to turn into a MAME cabinet. That hurt to watch, especially when he gave up on it and stuffed it out in the garage for a couple years. Luckily, it was a badly done JAMMA to MVS conversion, and the motherboard and carts were sold on eBay to presumably good homes. I wish I had a picture; the button placement was horrendous. They went for the diamond rather than the arch layout, but they came out more like misplaced, irregular trapezoids.
He eventually just let me have it. I took too long to get it out of the shed and he forgot he said I could have it. That was a fun argument. Luckily, my mother remembered him saying it and things calmed back down. But yeah, while emulation isn't my ideal route, I have to say that it made a pretty kickass way for a broke teenager to check out Neo Geo. Set up an extra computer with Linux and wahcade frontend, propped a computer monitor on a board resting on top of an inverted kitchen chair, and it made a kickass way to check out TSS, Metal Slug, etc. I'm sure I played them before that, but playing them on a crappy JAMMA to MVS to MAME converted cabinet is where I definitely, 100% remember starting to play more Neo Geo than I had before.