Find the closest palindromic number
Pyth, 26 20
Lnb_bWP`+QZ=Z-g0ZZ)Z

Updated to meet the new rules.
The program runs in an infinite loop which tests every possible increment, in the order 0, -1, 1, -2, -2 ...
Explanation:
Q=eval(input())     implicit
Z=0                 implicit
Lnb_b               def P(b): return b != rev(b)
WP`+QZ              while P(repr(Q+Z)):
=Z-g0ZZ             Z=(0&gt;=Z)-Z
)                   &lt;end while&gt;
Z                   print(Z)

Example run:
python3 pyth.py programs/palin.pyth &lt;&lt;&lt; 965376457643450
-2969881

This took 23 seconds.

Bonus solution, same character count:
Wn`+QZ_`+QZ=Z-g0ZZ)Z