I enjoyed my years with the Cube quite a bit (not as much as N64, but that was a better time in my life in general), and have had many memories with it. Here would be my top 10 (listed as chronologically as I can remember):
1. The night before launch my friend stayed over so we would be able to run to the store that morning to pick it up. My mom took us to Wal-Mart in a department store heavy area. Funny thing though; there were lines outside Toys R Us, Circuit City and Best Buy waiting for the stores to open (it was about 7AM). Meanwhile we walked into Wal-mart (which was open at 6), walked right to the electronic section, BAM, cabinet full of Cubes, and no lines whatsoever. I soooo wanted to drive back past the other stores and wave my Cube box like a madman out the window, but that would've been hazardous.

2. Everyone speaks of F-Zero GX extreme difficulty (and no doubt it was huckin fard), but the first game to push my frustration to the max was Wave Race Blue Storm. Battling the cheap AI on those frustrating waves in the harder difficulties had me chucking my controller often. Second would be that mission where you had to fight a 3 on 1 on Majora's mask in SSBM. Goes right up there as well.
3. Eternal Darkness was an experience like no other. The first horror game I ever played through in full. Of course I played the skillful (read: wimpy) way and avoided most of the sanity effects.

4. Metroid Prime. Plain and simple, just pure immersive fun. Totally ate away all my free time that Thanksgiving weekend. The battle with Prime will forever be etched in my memory of all time favorite boss battles.
5. Super Monkey Ball 1 was a superior single player game, but SMB2 was by far the better multiplayer game. Tons of hours spent playing Monkey Bowling with friends. Not in rotation anymore, but the memory will always be there.
6. No mention of this yet....probably the most explosive feeling ever during the Cube era for me was seeing the unveiling of the Twilight Princess trailer on a live feed at E3 '04. Holy jazeebus, the game you least expected has actually arrived!
and still arriving.
7. Finally getting a high praised 3D fighting game in Soul Calibur II on a Nintendo system. With Link to boot. I put mad hours into this game and still only achieved an intermediate level of play, which I quickly learned after participating in a tournament and getting spanked in the second round. But I was spanked by the CHAMPION, so my ego wasn't too hurt.

8. You'll notice a glaring lack of SSBM praise. Well that's because when you have friends who hate the game because they removed the cheap overpowered throws from SSB, playing alone becomes about as fun as throwing a tennis ball against your wall for several hours. So after burning out on SCII, Naruto 3 became the ultimate party game around here. Really, the most standout 4 player game this gen for me. Was my first import for the GC, and I'm glad things went smoothly.
9. I agree with ShyGuy on this one; RE4 was THE game to wave in the face of fanboys everywhere as a kiddeh image breaker. RE4 was not a game I was going to buy due to my lack of interest in the RE series and survival horror in general, but the promised new style of play and the eventual hype won me over, and I couldn't be happier. I'm often far too busy these days to replay single player games, but this one turned out to be the exception.
10. A pretty surprising moment for me here. A game that I bought on hype alone, and hadn't even opened it for a whole YEAR, just out of eventual boredom I popped in Tales Of Symphonia. RPG's really aren't my thing, as I've only played and enjoyed Super Mario RPG, Paper Mario and the Golden Sun series on GBA. But ToS really won me over, being a traditional style RPG with an untraditional battle system (though traditional for the Tales series). I hadn't really grown attached to a group of video game characters since, well, I can't even remember. Hopefully Namco gives us another Tales down the road.