Chain slipping off a non geared bike
So, chains aren't very different from tires in the sense that, they have to be replaced regularly, especially when you are commuting regularly. During summer and fall when I commute and race the most it's not unusual for me to replace my chain every other month. Chains get worn out, they stretch, they start to skip and behave poorly. If you haven't replaced that chain in two years, I'll bet my hat that it's falling off because it needs a replacement.

A new axle couldn't hurt either if you're axle is looking worn and starting to slip. You might have to trust the word of the repairman that pointed out the worn axle.