Yes. I'm aware of them. But that's not quite the question here. The question was whether there were equations that have solutions, that can be found, but where there's no algorithm to do so. So by definition the numbers of interest are definable in some way more succinct than their infinite representation.

As a nitpick: My understanding is that algebraic just means a number is the root of some polynomial equation with rational coefficients. But that's not the same as "cannot be described except with an infinite decimal string." A number might be described succinctly while still being transcendental. The natural logarithm base is transcendental -- not algebraic -- but can be described succinctly. It has a one-letter name ("e") and a well known series expression that converges quickly. I think you want to refer to something like the non-computable reals, not the non-algebraic reals.