Did the Earth once have 2 or more Moons?

It's thought that it did for awhile...

The early Earth was impacted by a Mars-sized world (called 'Theia'), ejecting trillions of tons of mass, that went into orbit about the Earth. This debris formed a ring of material that collected (rather rapidly) into a number of larger bodies, that in turn themselves crashed and stuck together.

At some point, there may have been *two* moons in our sky. As they revolved about the Earth, every time they were close in their individual orbits, the two bodies drew closer together - until finally "The Big Smack" happened. The evidence that suggests this is the highlands on the far side of the moon, permanently turned away from Earth which has mountainous landscapes very different from the smooth face visible on Earth. It's believed that those mountains are actually the remains of the smaller moon, which would have been around one-thirtieth the size of the larger body, that impacted the larger moon's surface.

Its certainly a possibility. There is a theory that during the Earth's early history, around 4.5 billion years ago, there were two moons.

The Moon was formed by the impact of a planetary object, often called Theia, with the Earth, which huge ring molten rock and dust around the Earth.

One fairly recent theory suggests that this cloud of debris formed two moons: a closer larger one and a further smaller, denser one. This second moon then collided into the back of the larger moon. The impact caused the whole moon the become semi molten again, and the matter from the second moon was effectively absorbed by the far side of our moon. This is used to explain the very different topography of the Moon's far side.

Start with the Wikipedia entry linked.

Because of the similar compositions of the moon and Earth, the hypothesis of a Mars sized body colliding with the Earth may be incorrect.

A more recent hypothesis:

To help explain problems with this, a new theory published in late 2012 posits two bodies five-times the size of Mars collided, then re-collided, forming a large disc of debris that eventually formed the Earth and Moon.[1] The paper was called “Forming a Moon with an Earth-like composition via a Giant Impact,” by R.M Canup.[1]

A late 2012 study on the depletion of zinc isotopes on the Moon, supported "the giant impact origin for the Earth and moon".[11]

In 2013, a study was released that indicated water in lunar magma was 'indistinguishable' from carbonaceous chondrites and nearly the same as Earth's, based on the composition of isotopes.[12][13]

GIH theory was again challenged in September 2013,[14] with a growing sense that lunar origins are more complicated.[15]

From that brief discussion, it appears that there is no agreed upon theory at this time.

Some computer simulations show that the collision that formed the moon could have easily formed two smaller moons instead of the one big one we have now. The simulation showed their gravity would have pulled the two smaller moons together, causing them to collide and merge, less than a million years after they originally formed. These computer simulations are the only evidence of a second moon we have, since the surface of the moon has been so well pulverized by asteroid impacts over the last 4.3 billion years.

I don't think there is anyway to know for sure, but most theories about how we got the moon involve tiny fragments = millions of tiny moons that orbited Earth like the rings of Saturn, before they combined into the present moon. Also there are likely thousands of grain of sand size moons presently orbiting Earth, temporarily = up to millions of years.

We have records in tidal rhythmites that go back a bit over 2.2 billion years, and for that period of time there was only 1 Moon.

One of the objectives of the NASA GRAIL mission was to collect data to confirm or refute that.