One can give literally hundreds of examples from classical analysis. usually there are 4 stages:
a) non-conscructive proof that some universal constant exists, usually by compactness arguments (in classical function theory this is called normal families argument).
b) a proof which is constructive "in principle", and gives SOME numerical estimate. Sometimes this estimate is ridiculoisly large (or small), and the author does not even care to write it.
c) Obtaining some reasonable estimate.
On this stage, sometimes a competition starts for better and better estimates, until
d) the exact constant is sometimes obtained. Which usually involves solving a variational problem and description of extremal configuration.
Some stages can be skipped, of course.
Some famous examples include "K\"obe constant", which turned out to be 1/4. Bloch's and Landau constants (they are almost century old, and currently they are on stage c)).
But literally hundreds examples from complex analysis can be given. Usually
Often the time between a) and b) is small. The largest time usually passes from c) to d), and many problems stop on stage c).
Here are two examples from my own work: arXiv:math/0607743, the time lag between a) and b) was 65 years. arXiv:math/0510502, a problem still in stage a)
The examples from analysis and number theory are so abundant that I don't think it is possible to catalog them.
