I see what you are getting at but unbounded utility is not needed for the result as the following counterexample shows.

Consider the utility function u(x1,x2) = 1 - 1/(x1x2) which is continuous, strictly increasing in x1 and x2, bounded above by 1 and has expenditure function e(p1,p2,u) = 2*sqrt[p1p2/(1-u)]. This is unbounded above as u tends to 1.

To answer the question, consider what would happen if expenditure were bounded above and the consumer were given an income above this bound. Some portion of income would not be spent but this is impossible as we know the consumer always wants more.

To put it formally, suppose for p>>0 that there exists a y* such that e(p,u)<= y* for all u. Now choose y** > y* and denote the maximum utility achievable with income y** by u**. Since the utility function is continuous and strictly increasing, by Theorem 1.8 we have e(p,u**) = e(p,v(p,y**)) = y** <= y* < y**, a contradiction.