The last solo trip I took, I rode ToT four times in a row. Same with Splash Mountain & Space Mountain. Most people I go with (boyfriend, best friend, running friends) only like to go on those rides once, maybe twice, but I find I never get sick of them. I like to ride ToT and Splash Mountain at least once during the daytime, and once at night, because they're different to me during those times.
I also do a LOT of shopping when I'm alone. I like to find little treasures that remind me of the Disney magic (pretty jewelry, interesting pins, souvenir t-shirts that I normally wouldn't waste time trying on if I'm not alone). When I'm with other people, I find myself enjoying watching the other person experience stuff more than I enjoy doing stuff for myself, so I shop less when others are with me.
And this may sound weird, but I really enjoy people-watching. If my feet are worn out or my legs are getting sore, I'll grab a drink or snack, find a place to sit down, and watch the reaction of people getting off the rides or at character spots.