How deep does a counter electrode need to penetrate the solution in electrochemical experiments?
TL;DR As long as you are not getting overloads on your instrument, it is submerged enough. 

Longer answer:
Your potentiostat will increase the potential on your counter electrode until it achieves the desired potential across the working electrode(WE) reference electrode(RE) circuit. The exact potential applied on the counter will depend on a lot of factors like solution resistance, amount of current flowing on the working at the potential desired, material your counter is made of and finally, the area of the counter electrode. 

If you end up with a situation where your potentiostat hits its electronic limits before the potential difference WE-RE is the desired value, you will start seeing overloads. 

If you are curious, you can always put a voltmeter between your WE and your counter electrode to see how much of a potential your instrument applies.