In fact, there do exist probability distributions without any (finite) average. Suppose, for example, that there were a 1/2 chance of the job staying up for one day, a 1/4 chance of it staying up for two, a 1/8 chance of it staying up for four, and in general a 1/2ⁿ⁺¹ chance of it staying up for 2ⁿ days for all integer n ≥ 0. If you try to work out the expected average length of time for one to stay there, you'll find you end up with infinity – and yet it's a perfectly meaningful probability distribution.

Not that I'm suggesting that AQA's job adverts do in fact follow this pattern or anything like it, but it is at least possible in theory :-)