I try to save everything I make, which I know I shouldn't, but I spend so long on each piece it's hard not to. Usually if I mess something up now it evolves into another form. I was throwing a big vase and I leaned over and knocked the rim. I had to cut off a good 5 inches which would have come in at the top. I could not get the remaining pot to narrow in at all so I left it, cut a hole in the bottom and now it's a large planter.
Altered rims are fun too and something I'm trying to improve on, even when I don't have to.