Not really, people could just purchase small physical commodities to use as a medium of exchange instead (which is how the concept of money began in the first place). No economy will ever go 100% digital precisely because not everyone will want every transaction to be recorded...
I use cash as much as possible, love the anonymous aspect. I hate the spam and tracking that follows non-cash transactions - especially from Uncle Sam.
One problem with a cashless society - it's hard to stash a wad of money in a jar and hide it. What are you going to do, stuff a bank book in a jar (and the state confiscates the account in the meantime). I suppose I could purchase jewelry and stuff it in jars ("small pysical commodities").