How many three man cell ninja teams are there in naruto?

Many. Lets count some:

1. Naruto, Sakura, Sasuke.
2. Hinata, Kiba, Shino.
3. Neji, Lee, Tenten.
4. Ino, Shikamaru, Chouji.
5. Tsunade, Jiraya, Orochimaru.
6. Sarutobi (3rd Hokage), Koharu and Omora (the advisors)
7. Minato (4th Hokage), and 2 unnamed teammates. *
8. kakashi, obito and rin

There're probably more that I cannot remember right now.

every ninja excluding root/foundation/ne or what ever u wnt to call it starts of as a genin team so will be on a team of 3 unless in some casses when a ninja has died quite being a ninja but they usialy get a replace ment or when they promoted then the amount on a team changes but there never realy is a 3 man team wept the sanin or if their sensei died as theirs the 3 genin AND their jonin sensei*