Are there sequences in which you can prove there are no closed form?

Question

Any finite sequences can be expressed as a polynomial, but there are many infinite sequences for which we have found no closed form. Is it possible that no closed form exists? Are there sequences in which we've been able to PROVE no closed form exists?

Answer 1

There are infinite sequences without closed forms, for any reasonable definition of "closed form". There are only countably many closed form expressions and uncountably many sequences.

Answer 2

As has been noted, no answer can be given until you define "closed form". You might, however, be interested in the sequence of busy beaver numbers, which is known to grow faster than any computable function (and thus can't be given by any computable function). I'm sure some people would happily accept that as an example for which there is, provably, no closed form.