Output a non-Context-Free Language
Python 2 (24) (26)

s="1"
while 1:print s;s+=s


Prints the language of strings in the symbol 1 whose length is a power of 2, i.e {"1"*(2**n) | n&gt;=0}. The s+=s is equivalent to s*=2; it doubles the string each time. 

Since unary (one-symbol) CFG's are regular, and unary regular CFG's are periodic past a point, this cannot be regular and thus is not a CFG. Alternatively, use the pumping lemma to see that the possible lengths of words in a CFL must have finite density and can't have arbitrarily large gaps.

Unfortunately, Python doesn't have a short way to produce an infinite counter. I'm sure some language can do for i in Naturals:print("1"*i*i) shorter (squares suffice just as well).

Edit: Can shorten to

s=2
while 1:print s;s*=s


which prints numbers of the form 2**(2**n), which can be taken as strings over 0123456789. Because the decimal length doubles at each step, this can't be context-free because the length gaps cannot satisfy the pumping lemma.