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The following is based on my research via google search and a calculator. Not all information may be accurate.
Of the seven planets in our solar system, there are 285 moons orbiting in total.
Spoiler
Venus: 0
Earth: 1
Mars: 2
Jupiter: 95
Saturn: 146
Uranus: 27
Neptune: 14
Including Pluto, 290. But if you count satellites orbiting smaller objects, such as other dwarf planet moons and asteroids, then that number jumps to 752. That we know of. There could be more.
And that's for one star system.
Of the milky way, there are about 100 thousand million stars. (100,000,000) If we assume there are roughly the same amount of moons per star system, that's 752 moons times 100,000,000 stars, which equals 752,000,000.
But there are many more galaxies than just the milky way. Estimates say that there are roughly two trillion galaxies in the observable universe alone. 752 moons per star system times 100,000,000 stars per galaxy times 2,000,000,000,000 galaxies equals 1.504 times 10 to the 23rd power, in scientific notation. (1.504*10^23) Written out, this number is 150400000000000000000000. That is one hundred fifty sextillion four hundred quintillion. That's... a lot of moons.
But there could be more galaxies outside the observable universe. Tens of millions of billions of trillions... of googols (1 with 100 zeroes.) Or, just 2. We don't know. There could be an infinite number of stars, of galaxies, of planets, of moons out there, or maybe it's finite. We just don't know what's outside the observable universe, or what isn't.