Just to reinforce: showing that e and pi are transcendental is hard. People had been essentially trying to find out whether pi could be represented algebreically or whether it transcended algebra (hence the name) for 2,500 years before Lindemann came along in 1882.
Hermite showed that e was transcendental in 1873 after a more moderate mere 100 years of attempt. It took him many years of his life and after doing so he simply refused to attempt pi, claiming that he couldn’t face the task! However Lindemann actually used Hermite’s methods when tackling pi, the task taking a fairly modest 9 years(!)
There are a few numbers that to this day have not been proved to be transcendental, though are believed to be. These include:[ul][li]Euler’s constant, gamma (the limt as n goes to infinity of sum(1/n) - ln(n))[/li][li]Catalan’s constant (the sum of -1[sup]k[/sup]/(2k+1)[sup]2[/sup] = 1 - 1/9 + 1/25 - 1/49 + …); and [/li][li]pi[sup]e[/sup][/ul]The fact that we still can’t prove the transcendentality of these numbers should give you an idea of just how hard the task is![/li]
Other numbers that we have proved to be transcendental include:[ul][li]Chaitin’s constant (the probability that a random algorithm halts);[/li][li]e[sup]pi[/sup]; and[/li][li]2[sup]sqrt(2)[/sup][/ul]The last is an example of the Gelfond-Schneider theorem mentioned by Jabba but has a special name (Hilbert’s number) because proving its transcendality was one of Hibert’s famous problems.[/li]
One last thing: although proving a number is transcendental is generally hard, it can be shown that transcendental numbers have the same cardinality as the reals, which means there are more of them than there are rational numbers. How’s about that?
I have a thing about transcendental numbers. Can you tell? 
pan