According to this article on the European Space Agency web site just after the Big Bang and before inflation the currently observable universe was the size of a coin. One millionth of a second later the universe was the size of the Solar System, which is an expansion much much faster than speed of light. Can space expand with unlimited speed?
Answer
Yes, the expansion of space itself is allowed to exceed the speed-of-light limit because the speed-of-light limit only applies to regions where special relativity – a description of the spacetime as a flat geometry – applies. In the context of cosmology, especially a very fast expansion, special relativity doesn't apply because the curvature of the spacetime is large and essential.
The expansion of space makes the relative speed between two places/galaxies scale like $v=Hd$ where $H$ is the Hubble constant and $d$ is the distance. When this $v$ exceeds $c$, it means that the two places/galaxies are "behind the horizons of one another" so they can't observer each other anytime soon. But they're still allowed to exist.
In quantum gravity i.e. string theory, there may exist limits on the acceleration of the expansion but the relevant maximum acceleration is extreme – Planckian – and doesn't invalidate any process we know, not even those in cosmic inflation.
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