How many bits of information can be stored in an atom? The atom in question being as big as you like, but must be stable with regard to nuclear decay
I came here thinking information as something what could build the atom. Reading here I noticed the terms come from different disciplines, so I googled for Information Theory and Physics and got this: https://en.wikipedia.org/wiki/Physical_information
Found what is a bit related do the physics entropy:
The base of the logarithm used in this definition is arbitrary, since it affects the result by only a multiplicative constant, which determines the unit of information that is implied. If the log is taken base 2, the unit of information is the binary digit or bit (so named by John Tukey); if we use a natural logarithm instead, we might call the resulting unit the "nat." In magnitude, a nat is apparently identical to Boltzmann's constant k or the ideal gas constant R, although these particular quantities are usually reserved to measure physical information that happens to be entropy, and that are expressed in physical units such as joules per kelvin, or kilocalories per mole-kelvin.
So I conclude an atom, representing something to physical world, is related to a bit if the math of your physical system model brings a log of base 2.
Your question then depends of the system what you're observing.
According to Planck one proton can contain a thousand - trillion - trillion - trillion - trillion - trillion bits of information. This is the area (not volume) that a black hole expands when you add one bit of information. Information is not lost but information can not escape a black hole. When you erase your hard drive the information still exists as energy. (Leonard Susskind) https://www.youtube.com/watch?v=2DIl3Hfh9tY