I first saw a IIe in a computer store while on a family trip when I was in about 8th grade. I begged my parents to stop there and remember playing a demo that had a rabbit moving around a maze to get a carrot. I begged my parents incessantly to get one and eventually they caved, buying one from a local Team Electronics store. I had a few games, but couldn't afford much so I played around with BASIC programming. Once I got my 300 baud Hayes MicroModem II and discovered pirate BBS's and AE Lines, my game library increased exponentially! We also got IIe's in our high school and traded a ton of games there.
For me, the IIe was all about getting and playing games, and Broderbund games were my favorite - Karateka, Prince of Persia, and Wings of Fury. AutoDuel by Origin Systems was also a favorite. For two player games, I liked Hardball and Mario Bros, which of course my friend and I only played in deathmatch mode!
I ran a BBS / AE Line of my own on and off for several years and revived it a couple of years ago, so I'm having a blast reliving my Apple II days. And it's so cool that all these great games are now being converted to ProDOS by qkumba so we can play massive collections of games, like Total Replay, right from our hard drives!