How many atoms are there in our universe?

Here is one way to do this calculation.

The horizon size of our universe is about 14 billion light years. If we neglect space curvature effects, the volume of visible space represents 4/3 pi R^3 = 8.8 x 10^83 cubic centimeters.

The critical density of the universe for this value of the Hubble constant is 3 H^2/8 pi G, which works out to be 1.0 x 10^-29 grams/cubic centimeter or about 0.000005 atoms of hydrogen/cc. It is believed that only 4 percent of the critical density is in the form of normal atoms, so this leaves 0.000005 x 0.04 = 0.0000002 hydrogen atoms/cc. Multiplying this by the volume of the visible universe, you get about 1.7 x 10^77 hydrogen atoms.

In the TOTAL universe, including the part outside our horizon, the answer is