Generate an aperiodic integer sequence
Golfscript, 2 characters
`,

(stringify (`) the number and count (,) the digits)
J, 3 characters
#q:

is the count (#) of the prime factors (q:) of its input.
The vector form involves more bookkeeping, but that's not our contestant:
#@q:L:0&amp;.&lt;

the first 40 terms are: 0 1 1 2 1 2 1 3 2 2 1 3 1 2 2 4 1 3 1 3 2 2 1 4 2 2 3 3 1 3 1 5 2 2 2 4 1 2 2 4
see more at OEIS
The sequence is 1 at every prime position, 2 at every prime multiple of a prime, 3 at every prime multiple of a product of two primes...

If you are not convinced this is aperiodic, here are plenty of other sequences:
##:

(the number of bits = floor of the base-2 logarithm)
+/#:

(count of 1-bits in the binary representation of the number). This (reduced under any modulus)
#b#:
+/b#:

(the same in base-b)
&lt;.^.
&gt;.^.
&lt;.b^.
&gt;.b^.

(floor or ceiling of the (natural) log - this time without stringification)
*./#:
+./b#:
*./b#:

(GCD or LCM of all base-b digits (except GCD of bits won't work))
1{#:!
9{#:!
({#:@!)

(second/tenth/nth highest bit of the factorial of n)
(=&lt;.&amp;.%:)

(Is the number equal its floor under the square root operation? Is the number equal the square of the floor of its square root? Is the number a square?)