What word would you use to describe someone who tries to one up everything you say?

What word is appropriate for someone who tries to one up everything you say? For example:

Me: "You are a great friend"
Person: "Well, you are my best friend"

As implied by the question itself, the standard term for the behaviour itself is one-upmanship.

Although alphadictionary.com is prepared to accept one-upman as a word in itself, to be honest I think they're in the minority on that one. In the absence of a convenient single-word derivation from the standard term, people usually just say it using more words - for example...

He always has to one-up everything you say. (lots of written instances in that link)

Most of us indulge in one-upmanship from time to time - it's part of normal social interaction. It only becomes tiresome with those few people who do it excessively. The most common single-word adjective I can think of for such a person is competitive. Which has lots of other shades of meaning, but when used in the context of how someone interacts socially, I think most people will know exactly what you mean.

user653's comment to @Barrie's answer reminds me of my favourite example from Dilbert...

I've remembered the word I was trying to think of earlier. It's topper, although it's not in the OED as such.