Suggesting closed-form representations of mathematical constants by means of experimental mathematics? Quite a  few mathematical constants are known to arise in several branches of mathematics (more here). I have no doubt that they are useful to the understanding of some mathematical structures and add to the whole body of knowledge of mathematics as a whole. Sometimes, however, I find myself looking at these constants and I feel unsatisfied. I suspect that more constants can be expressed in more 'fundamental' constants like $\pi, e$ and $\gamma$.
Although Euler probably came close to be a living one, mathematicians of the past did not have computers. We can now use computers to determine te value of the afore mentioned constants with great precision. We can also try to conjecture the exact value of these constants by finding the value of some arithmetic combination of $\pi, e$ and $\gamma$ and the real numbers that corresponds to one of the 'unevaluated' constants with high precision. My question is: have experimental mathematicians 'found' the value(s) of some of the constants I described (by the the method I described or a similar one)?   
Thanks,
Max
P.S. If someone knows of a reference to a paper/book that summarizes some results in this subield of a field, I would be grateful to him/her.
 A: Have you tried the Inverse Symbolic Calculator?
A: The rational constant for an Apery like series for $$\zeta(4)$$ was found experimentally, using continued fractions.
A: I can't resist giving a couple of examples of (contrived) numerical coincidences (both are from Experimentation in Mathematics by J.M. Borwein, D.H. Bailey and R. Girgensohn).
Example 1.
$$\int_{0}^{\infty}\cos(2x)\prod_{k=1}^{\infty}\cos(x/k)\ dx=\frac{\pi}{8}-\epsilon,$$
where 
$0<\epsilon<10^{-41}.$
Example 2.
$$\sum\limits_{k=1}^{\infty}e^{-(k/10)^2}\approx5\sqrt\pi-\frac{1}{2}=8.362269254527580...$$ 
Well, they agree through 427 (four hundred twenty seven) digits yet they are not equal.

A moral. Make sure you understand the context and use your inverse symbolic calculator with caution.
A: Gauss discovered the AGM/elliptic integral connection through the value $$\frac{\omega}{\pi}$$ by experimental mathematics.
A: Not many people know that WA is also able of giving you all kinds of possible closed forms concerning fundamental constants, e.g.:
http://www.wolframalpha.com/input/?i=2.9299372410244
You can also click "more"...
A: I had a math teacher whose hobby was collecting "first fail" examples.   Some of these were conjectures that were true for hundreds of cases before a counterexample was found.  Matching of numerical constants to many digits should never be used as a proof; rather it should stimulate the hunt for a proof.
