From Berlin, the closest non Schengen destination is the Kaliningrad Oblast. The beeline distance is +- 500 kilometers and over the road its +- 600 kilometers. There are direct flights between Berlin and Kaliningrad. It is just one hour.
Otherwise, there are a couple of places in non Schengen countries that you can reach in more or less 2h30 by plane: Sofia (Bulgaria), Bucharest (Romania), Belgrade (Serbia) and Zagreb (Croatia). So as to find the most cost effective option, I would play a bit with Matrix Airfare Search or a similar tool.
In 2 hours you can also reach Kiev (Ukraine) or Minsk (Belarus) from Berlin. However prices for direct flights to these destinations are much higher than for the other destinations mentioned.
Depending on your citizenship, all these destinations may not be visa free. And a visa may not be obtained for free. Take that into account.