One example is indeed that of Transcendental numbers, as Yoav Kallus points out in the comments.

Liouville showed in 1844 that numbers which do not satisfy a polynomial equation with integer coefficients exist, but he only gave an example in 1851, the famed Liouville constant, a celebrity among Transcendental numbers:

$$\sum_{n=1}^\infty 10^{-n!}.$$ For more information, see here and here. You might need JSTOR access to read the first.

Cantor however, whose proof of the existence of Transcendental numbers follows directly from the uncountability of the Reals, only came up with his proof in 1874. Whether his proof of the Uncountability of the Reals is constructive or not is something people are still debating, so I will not comment on that. For more information on this, see this Wikipedia Article

One example is indeed that of Transcendental numbers, as Yoav Kallus points out in the comments.

Liouville showed in 1844 that numbers which do not satisfy a polynomial equation with integer coefficients exist, but he only gave an example in 1851, the famed Liouville constant, a celebrity among Transcendental numbers:

$$\sum_{n=1}^\infty 10^{-n!}.$$ For more information, see here and here. You might need JSTOR access to read the first.

Cantor however, whose proof of the existence of Transcendental numbers follows directly from the uncountability of the Reals, only came up with his proof in 1874. Whether his proof of the Uncountability of the Reals is constructive or not is something people are still debating, so I will not comment on that. For more information on this, see this Wikipedia Article