One can get a good estimate of the number of atoms in the Earth by taking its mass and dividing it by the each of its major constituents.

We don’t need to take into account the atmosphere, water, us, in the equation, as all those things, indeed, the whole biosphere, are but a thin shell on a much more massive world. They are a “rounding error” in the estimate.

The mass of the Earth is roughly 6 x 10^27 grams

The composition of the Earth is about 32% iron, 30% oxygen, 15% magnesium, 13.9% sulfur, 3% nickel, 2% calcium, 1.4% aluminum by mass. There are dozens of other elements in the Earth’s crust, but, since we are dealing with rough estimates, and they amount to less than 1% of the total mass, we can effectively discount them.

So that means that there is:

1.9 x 10^27 g of Iron

1.8 x 10^27 g of Oxygen

9.0 x 10^26 g of magnesium

8.3 x 10^26 g of sulfur

1.8 x 10^25 g of nickel

1.2 x 10^25 g of calcium

8.4x10^24 g of aluminum

Because we know the atomic mass of each element, we can figure out how many atoms of each are contained in a sample of a given size. The atomic mass is the weight of 6.022 x 10^23 atoms of that element in grams. 6.022 x 10^23 is a quantity of material called a mol.

1 mol of:

Iron: weighs 55.8 grams

Oxygen: 16.0

magnesium: 24.3

sulfur: 32.1

nickel: 58.7

calcium: 40.1

aluminum: 27.0

So, we can divide to determine how many mols of each element there are:

3.4 x 10^25 mols of Iron

oxygen: 1.1 x 10^26 mols

magnesium: 3.7 x 10^25

sufur 2.6 x 10^25

nickel: 3.1 x 10^24

calcium: 3.0 x 10^24

aluminium 3.1 x 10^24

all all those mols up and you get a total of: 2.16 x 10^26 mols

Multiply that by 6.022 x 10^23 and you get:

1.3 x 10^50 atoms

or, to put it another way, roughly:

130,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 atoms