I once got as far as downloading the application for for The Great British Bake Off, and kept it on my desktop for about a week before putting it in the recycling bin. I would never consider going on any other show though.
Yes! The 1900 House. If you haven't heard of it, it's about a modern family who live for 3 months in a modified Victorian style house. They have to dress and live as they would have back then. I love history and think it would be awesome to see what life was like around that era.
I would not apply for any type of exhibitionist reality show like a top model (even if I had the look, the shape and was the right sex) or Got Talent and so on. I would however apply to be on a renovation show, assuming I fit the criteria.