Depends. What's a country? Very hard to define, it turns out. The CIA World Factbook has a list, and some people go by the UN list of member states. I believe that's the list Chris Guillibeau goes by (he's visited all of them).
If you're competing, and want to record it, Most Travelled People and Traveller's Century Club are the two main ones that I found when I got a bit competitive at one stage - I wanted to reach 50 countries by age 30, and ran into the same difficulties.
However, you start realising that everyone has a different view on what makes a country. Is Kosovo a country? I'd say so, people who live there say so, but the UN doesn't (last time I checked). Is Scotland? Some say yes, some say no. What about Sealand? Micronations that exist only one day a year, like Uzipis? It gets very tricky.
And if you're 'competing', the problem with MTP and others is that 1) you have to register, and 2) you have to pay. I'm not paying to be on an arbitrary brag list. Some do, clearly, and indeed, a friend is on MTP and claims to be the x most travelled NZer, even when he joined I'd been to more countries (he's since passed me, but that's besides the point).
In the end, it's best to make your own definition. My friend has as well, he counts what he calls Tier 1, 2 and 3 countries (depending on their independent status and so forth).
Don't forget, and this is what got to me - if you 'visit' a country, have you actually seen anything? I went to Paris for a few hours once, but I decided not to count it as I only saw the airport and a hotel. I've since been back. It's more important to experience the people, the food, the culture, the sights, sounds and memories than a number. However, I get it, I'm also driven by numbers, and I list them in my profile.
So what you're going to have to do is either get your friend to agree with you on one of those 'definitive' lists, and then use that as your reference. Let us know what you choose, and good luck! :)