Invasion until the limits, if any

During the first steps of evolution (biological) species try to propagate using reproduction and taking resources. This strategy will not go too far. A proper AI would use exponential learning to replicate itself and improve faster and faster.

Eventually it could expand along the whole universe. Could be that AI develop the capacity to create new universes using simulation and then expand itself inside those universes to take their resources. These simulated universes could be able to create more universes inside and so on.