There are the big three: Hades, Poseidon, Zeus. Furthermore, the Fates, the Furies, the judges of the Underworld, the elder Cyclopes, the Hecatoncheires and the Oneiroi.
There are also three rules of the universe: the Primordials, the Titans, and the Olympians.
I am writing a book and three seems to be a theme in Greek mythology. Any other groups of three would be helpful.
Definitively answering the question of "how many" would require an enormous research effort, and it's a little fuzzy in terms of what qualifies. Being lazy, I only say triads are important and quite common.
I suggest taking a look at:
It lists manifestations such as the Boeotian three muses configuration (Melete, Aoide and Mneme, which legendarily pre-dates the nine muses.
Maiden/Mother/Crone configurations are particularly prevalent, for instance Kore/Demeter/Rhea. (The prevalence of female triads relates to the three distinct biological conditions related to reproduction--pre-pubescent, fertile, and menopausal. Pagan religions tend to be preoccupied with cycles of generation.)
Among the most famous triads are:
More fearsome famous triads are:
And of course:
A famous tri-partite male figure is
Although it's important to recognize his connection to Demeter and the Eleusinian Mysteries.
Is an important triad, where the fates of the three warriors are entwined.
In this famous version of the riddle, "the creature that goes on four legs in the morning, two legs in the day, and three legs at night" is an an analogy for the three phases of human life: childhood, adulthood, old age.
