This is a difficult one to answer!

As you didn't really specify a particular time-period, I'll try to be as general as possible.

It is difficult to come up with a precise number of gods for the following reasons:

A) There's no such thing as a unified Babylonian pantheon. Originally (i.e., in the third millennium BC, when our sources start), each city in southern Mesopotamia had its own patron deity and its own pantheon. These gods were, however, still venerated in other parts of Mesopotamia, but sometimes under a different name. In turn, there are specific local manifestations of certain deities, such as, for example, Ishtar of Uruk vs. Ishtar of Agade.

B) which gods were venerated changes over time (for example, some deities who were very important in the third millennium BC disappear from our record in the second and first millennium BC, whereas new deities appear and gain importance)

C) there is a fair degree of syncretism going on, meaning that different gods are equated with other gods, and the same god can have several names or epithets, so it's sometimes very difficult to know if its a different god or the same god with a different name.

D) there are Sumerian and Akkadian-language names attested for many gods.

This is just to say that what follows is more of a ballpark figure:

There are about 15 'greater gods' (although not all of them were of equal importance at any given point in time). The most important one in Babylonia in the second and third millennium is Marduk, the city-god of Babylon. Other important deities are Enlil (the head of the pantheon in the earlier periods), Ishtar (the goddess of war and sex), Ea (the god of wisdom and cunning), Shamash (the sun-god) etc.

In addition to the greater gods, there are hundreds of smaller deities. Some of them are servants of the greater gods or belong to their courts (just like humans, the greater gods have advisers, messengers and so forth). Others are simply regional gods or 'poor relations' of the great gods.

Sources / Further Reading: Very useful online resource on Ancient Mesopotamian Gods and Goddesses

The standard books on Mesopotamian religion are still: Jacobsen: The Treasures of Darkness. A History of Mesopotamian Religion (1976) and Bottéro: Religion in Ancient Mesopotamia (2001)