Greatest Common Divisor
Retina, 16
^(.+)\1* \1+$
$1

This doesn't use Euclid's algorithm at all - instead it finds the GCD using regex matching groups.
Try it online. - This example calculates GCD(8,12).
Input as 2 space-separated integers.  Note that the I/O is in unary.  If that is not acceptable, then we can do this:
Retina, 30
\d+
$*
^(.+)\1* \1+$
$1
1+
$.&amp;

Try it online.
As @MartinBüttner points out, this falls apart for large numbers (as is generally the case for anything unary).  At very a minimum, an input of INT_MAX will require allocation of a 2GB string.