Not really. Factorial practically begs for a recursive implementation and it's very simple.
Then there's fibonacci, qsort, etc.
Well, they can be done recursively, but their usual definitions imply the simpler iterative approach. Using them leads to the problem that you often see, not just with young students, but even with experienced "professional" programmers: They learn that recursion is just a complex, obscure way to do iteration.
If you actually want to get across why recursion is important, you really should use examples in which recursion gives a simpler solution than iteration. One of my favorites, partly because people are usually surprised to discover that it's actually best done recursively, is a task that software does a lot: Given a binary number, generate the decimal representation of the number. The natural (iterative) divide by 10 and output the remainder of each step gives the digits in reverse order. This is fine if you're putting the result in a fixed-width field that you know is wide enough, but it's not fine if you're generating ordinary text with just one space before and after the number or if you don't know how many digits the number will have. To generate the number iteratively in the order we usually say or write the digits requires two passes, one to count the digits, and the other to write them. Or you can generate the digits in little-endian order into a large buffer, then use a second iteration through that buffer to output the digits in big-endian order.
But a faster, more elegant way is to write it recursively, with a routine that saves its remainder digit while it passes the quotient to a recursive call of itself. The bottom-level call finds it has a 1-digit number so it doesn't make the recursive call, but simply outputs its digit, and returns to the caller, which writes the 2nd digit, and so on. Students that understand this now know that recursion can sometimes simplify some (but not all) problems.
There are number of other simple problems that are best solved recursively, but this page's margin is to small to hold the list. ;-)