Post-Transcendental Numbers

I might be wrong off the starting bat here, but e can be defined in an infinite series 1/0! + 1/1! + 1/2! + … + 1/n!

So, i was wondering, Does a number exist that:
-> Cannot be expressed as a fraction (Irrational)
-> Cannot be expressed as a polynomial (Transcendental)
-> Cannot be expressed by an infinite series following a set rule (???)

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