This already happened, if all the Universe started from a infinitely small point.
The Universe did not start from an infinitely small point. Inflationary Cosmology makes it clear that there was a universe before the big bang. The old theory of an infinitely small point with infinite density and infinite energy raised serious compatibility issues with the horizon temperature, nor does it explain why the big happened and where matter came from.
The big bang is the inflaton expanding the three enlarged spacial dimensions into the forms we recognize them today.
If the universe is expanding, continuing the big-bang inflation.. Isn't the size of the atoms and sub-atomic particles that make them up expanding too? And the space between them?
First off, the issue is the expansion is INCREASING. Expansion on its own is not that big of a deal, since the original equations of General Relativity suggest that the universe should be either expanding or contracting and that it is extremely unlikely to exists in a sort of equilibrium that would entail a static universe.
And the increase in the expansion rate is not the same inflationary expansion that was part of the big bang. That inflaton has gone and expansion slowed down, but at some point it appears that it has increased again. We have an explanation for the inflaton at the big bang as being the Higgs field, but figuring out why expansion is speeding up again is more than likely going to require a new explanation.
But yes, the size of atoms, and the space between does indeed grow, but its change is not noticeable nor even relevant. Spacial expansion can only really be seen on the largest of scales, between two objects massively separated in distance from each other. As it has been said earlier in this thread, the further apart two objects are, the faster they are flying apart due to the expansion of the universe, and this rate is increasing.
Fizyk brings up the good point on the forces of nature. Expansion of space is not a big enough force to overcome electromagnetism or the strong force, so atoms will not get ripped apart from the expansion. Nor is the expansion big enough at scale to overcome gravity's attraction, so no worry about galaxies getting ripped apart, rather picture galaxies flying apart from other galaxies, but within, the structure is intact.
But elementary particles don't have size? I thought the age of point-particles was over...it is time to embrace the Strings, which clearly says that the elementary components of the universe have definate size and patterns.