This is a question that has amused philosophers and psychologists for centuries. Some models assert that society is merely the collected sum of individuals, whereas others would argue that the individual's identity develops alongside and with reference to other individuals. You seem to be in favor of the latter, that one needs external relationships in order to develop as an individual.

So your question is "how many?" In the end, that's going to depend largely on each person's specific needs. I'll be on the extreme end of that spectrum, as I can go several weeks without any sort of human contact and be relatively fine, so long as I'm not bored, and forming a meaningful bond is damn near impossible. On the other end of the spectrum, you'll likely find people who'll show signs of depression in a matter of days if left in complete isolation.

Regardless, I would argue that quality is far more important than quantity, and that a couple of close relationships should suffice. I would even argue further that, if someone needs recognition from a great number of people in order to feel happy, they're likely missing the point.