I have one that I use when going out at night. I can lock it up in any neighborhood and not have to worry about it because if it's stolen, I haven't lost a ton of money on it. Also, it isn't a valuable or attractive bike, so I'm hoping it would be less of a target to bike thieves. In my case, my beater fits your first criterion, in that it's an old frame with a mix of old and some new parts. But it doesn't really fit your second criterion because I do maintain it and give it a good amount of attention, since I ride it quite a bit.

One important note is that I really like to ride it! Being a beater doesn't mean it can't be really enjoyable to ride. If it isn't, it's probably not going to be in my stable for long.