That really depends on how long you'd want to stay in each country & what you like to do. I'm currently traveling the world (6 months now and many more months to go hopefully!), but I'm a slow traveler, meaning I like to really delve into the history & culture of each city I go to, so I like to stay a minimum of 1 week in a city and I've stayed more than a month in certain countries. So, in 6 months, I've only been to 8 countries (14 cities). I know of someone else who traveled for 6 months and went to like 20+ countries, but she booked all tours which allows her only a few days or so per city. So, I guess you have to figure out what your travel style is. Plus, how much money you have saved up & which countries you plan to go to. Money (especially US Dollars) will last a REALLY long time in Southeast Asia, and a REALLY short time in Western Europe/UK (because of the exchange rate).
I would say around 60 in a year sounds reasonable.
260 if you do not stay very long in each one.
As much as you want to