Whenever I set about trying to learn new ProgrammingLanguages? (very frequently) I often find myself stuck without simple exercises to do. I hope that if people add neat simple programming exercises here, it could be quite useful.
"The oldest known nontrivial algorithm is Euclid's Algorithm for computing the greatest common divisor (gcd) of any two positive integers (300 B.C.). The idea of the algorithm is based om the observation that, gcd(a,b)=gcd(b,remainder(a,b)), and that gcd(x,0)=x."
Seems trivial? I agree, but it's a start.
http://moneyethics.com/Articles/compound.html
This usually a very trivial program, but a good first exercise when learning how to use a new programming environment (see HelloWorld).
Make your computer sing the 99 Bottles of Beer on the Wall song.
Very popular: http://99-bottles-of-beer.ls-la.net/
The TPK Algorithm is DonaldKnuths HelloWorld. It doesn't actually do anything but it has an array, loop, conditional and function in it to make it exciting.
See 
Wiki:WardNumberInManyProgrammingLanguages
A graph theory problem that looks convoluted enough to get very high-level languages sweating.
Wiki:EsotericProgrammingLanguages, like Brainf*ck, usually have small instruction sets and simple grammars. Interpreters for these languages aren't big programs but need a fair sample of a programming language's features to implement.
An example: befunge.a68 A Befunge interpreter in Algol 68 (the esoteric in the obscure).
GCC is written in C. The Oberon system is written in Oberon. Writting a Compiler in the language it compiles is a great way to make sure you understand the language. The hard bit, of course, is the BootStrap.
One page links to StuffToCode:
