There also is a counter used for hashing. So for a new password you just increment the counter. Remembering the counter for every site sounds too complicated, but you could store that in a file without losing much (any?) protection.
Wouldn't you want to maybe... encrypt that file? Seems almost circular unless you use something like stenography to embed the data somewhere. I'm not personally too thrilled by a counter file to replace a different file, at that point I feel like I'm losing features.
Yeah, but then you need to carry the file with you, opening it (especially on mobile) is super clunky, and when you get to a site that doesn't support numbers/letters/whatever that the generator uses, it's hell. I used to use that, but I switched to KeePass2 and it's much better.