I would pick Chaitin's constant.
It is not only transcendantal like Pi, it is even uncomputable, so it would be part of his counterexample set.
(Pi is computable, so you can describe it with an algorithm (finite number of characters))
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But can't Chaitin's constant be described using a finite set of symbols, those symbols being in the set [achinost'] (at least, in English)?
Sure; a simple definition is at https://tromp.github.io/cl/Binary_lambda_calculus.html#Halti...
OK, original poster doesn't speak about computable numbers: "consider the set of numbers that cannot be described using any finite collection of symbols"
But I have a hard time accepting numbers, which can only be described in natural language.