There is no limit, in theory or practice. This thread on Flyertalk mentions several people with five citizenships, posits a case where a child born to a couple with three citizenships each would easily get 7, and goes on to speculate that with the right sets of parents it would be possible to acquire more than 9 at birth plus any more you'd care to accumulate during your lifetime. There are still quite a few countries where you can become a citizen in less than 5 years, or even buy citizenship outright if you throw enough money at them.
Once past the age of 20 or so, simply maintaining a citizenship generally requires zero effort, once the bit is toggled it stays toggled short of (say) actively waging war against a country you're a citizen of.
However, citizenship does often incur obligations like tax, military service etc, which can make it a pain to stay in the good books of all your countries. As a simple example, if I'm a male Finnish citizen over 18 who hasn't completed their national service, I'll stay a citizen if I stay overseas, but I'll be arrested as a draft dodger if I return to the country.