Hi folks, I'm making a trivia game for an advanced high school math club.

I'm making a sequence-completing quiz (e.g. 1,1,2,3,5,8,?) but I'm looking for a few "gotchas", i.e. sequences that might go 1,2,3,? where ? ends up being 16,453 or something surprising.

Obviously any arbitrary sequence can be as surprising as I want, so I'm looking for non-trivial ones (in the sense that Graham's number[*] is the largest "non-trivial" number in math), even better if it links to some deep important theory that can be described, or if it would be instantly recognizable to a professional mathematician!

[* they learned up-arrow notation already, that was part of the inspiration for this quiz!].