Believe me, Crisgal - despite what Corylus writes above - 'cleverer' is always perfectly acceptable in British English. There are even circumstances in which "cleverer than me" is acceptable, too!
Someone might, for example, say, "The teacher thought him cleverer than me." It's clear that both personal pronouns must be in the accusative/objective case. This means that the teacher thought that person (him) was cleverer than he/she (the teacher) thought I (the speaker) was.
Had the person written, "The teacher thought him cleverer than I", that would have meant that the teacher thought that person (him) was cleverer than I (the speaker) thought he (that person) was. A completely different proposition, of course!
In any case, the so-called 'correctness' of 'I' in such citcumstances is very much in doubt, according to the actual usage of the language. Imagine you have told someone you will call on him later in the day and you arrive at his house and knock on the door. A voice from within says, "Who is it?" How do you respond? I can almost guarantee - unless you are an extremely pedantic professor of linguistics - that you will call out "It's me!" That professor would claim that the verb 'to be' does not take accusative forms after it and you should, therefore, have said, "It is I."
On that basis, I would claim that "He is cleverer than me" is perfectly OK in day-to-day usage.