I have 3: Born in Canada (so only nationality I got automatically), British from my father and Dutch from my mother. I lived enough in Italy to apply for Italian citizenship, but the question of National Service made me duck that one.

My nieces were born in France and so have four citizenships.

If I were to have kids with someone with 2 other citizenships, have them give birth in a country which provides citizenship upon birth and bring them up somewhere else, they'd have 7, which is probably getting a bit much.