Super Mario RPG: This game brought so many new features to Mario along with 2 new characters, a great plot (that is being overused) and can get somewhat challenging (try beating [color=000000]Culex</span> without any armor or weapons).
Age of Empires II: The Age of Kings: This is a time consuming RTS game that seems to have infinite possibilities. You can choose to fortify your self and try to slowly expand, try to bombard your opponents with legeons of soldiers or build up your economy until you can dececrate your foes.
Extreme Tux Racer: I never know where I am and yet I always can find my way. The herring are a nice addition and the options can help change the game. It has a certain feeling to it that makes it very fun.
Super Smash Bros. Melee: This had so many things that made it more exciting than Brawl. It had the fun bonuses and clever events that seemed much harder than those in Brawl. The Adventure mode was short but sweet. It took longer to complete than the random classic mode and had quite a few secrets and bonuses. The trophies were great fun to unlock, even if you didn't know what to do. You would try everything to get them. The fast paced battles would boost the usefulness of certain non-glitch techniques. Even with the glitches being exploited, it's still one of the best games I can think of. (The others also being included in this post.)