When do prices not exist?
There are a bunch of examples for incomplete markets in the finance literature.  The oldest (that I know of) is Hart (1975).  In finance, the problem is that if you have two different assets they have different prices that make the market complete.  But then equilibrium considerations make them have the same price, which forces the market to no longer be complete.  (If the market was complete, then they would have the same price.)