Friday I stopped by a bank to get some $2 bills to george. They had some so I was in luck. The teller said, "I've got some, but let me give you the clean ones." So I got 8 clean $2 bills. While she was separating out the messed up ones, I was watching for a WG stamp on any of them. I didn't see one. The ones she was separating out and not giving me appeared to be (a) washed such that the ink was very light (2) ripped in half and taped back together and (iii) 1/4 missing.
This made me think. Since $2 bills only leave banks when someone specifically asks for them, would a simple WG stamp cause the teller to put a georged $2 into the messed-up pile and not give out? It's not like a bill of another denomination that gets buried in a stack of other bills and is given back out with hardly a look.
I'm thinking the best strategy for $2 bills is to chicken george them with just a single stamp on the back.
So far I've entered 42 $2 bills and have a total of zero hits. I've stamped them both front (once) and back (twice).