How long would it take for an alien civilization to populate an entire galaxy?
Let's also say that they are slightly more advanced technologically than we are. So, they can build ships of some kind that either a) have AI that can self-replicate if they find a planet with materials to build more probes, or 2) carry aliens with them that can settle* a suitable planet they find. They then establish themselves, grow, and send out more ships when they are capable.
It's an interesting question. We want to know if We Are Alone, for example, and if we are being visited by aliens (spoiler: We are not). It's a big galaxy — the Milky Way is a flat disk about 120,000 light years across — and if you have relatively fast ships it still takes a long time to cross it.
But the premise involves exponential growth, so the spread of the settlements means more growth, which means more settlements, on and on. The aliens would spread across the galaxy in a wave. What would that look like? And how long would it take?
This problem has been tackled many times, but a recent paper by a team of astronomers (one of the authors is Jason Wright, about whom I've written many times before, like here, here, here, and here) looks at it a little differently. They make some interesting assumptions, and what they find is that even being really pessimistic the entire galaxy can be explored in less than 300 million years, far shorter than the galaxy's lifetime.
