Depending on various current estimates, our galaxy is thought to have somewhere in the region of 100-500 thousand million stars.
If we assume that a colonisation mission is launched to another nearby star system (any system - I'll explain why in a moment), and takes no more than 10,000 years to get there and set up a colony, then sends out 2 more colonisation missions to other nearby systems, then in the course of 1 million years, the number of colonised star systems would equal approximately 2 to the 100th power - or 1.3 million million million million million - star systems. This number is far greater than any estimates of the number of stars (and hence star systems) in the galaxy, which strongly implies that our descendant would have completely colonised the galaxy by that point in time.
Now, why did I say that it doesn't matter which systems we travel to? Because I am including in this the implicit assumption that by this distant point in time, we will have perfected terraforming capabilities to make planets habitable for the colonists, or else altered the colonists so that they will be capable of living and thriving on their destination worlds. Such altered genetically-altered colonists would still count as "Humanity or its descendants" for the purposes of this bet.
As to the travel time between star systems, I have assumed that the average distance between stars (all being assumed to have at least 1 planet for simplicity's sake) is probably in the range of 5-10 lightyears throughout the galaxy (although in the galactic core, it would no doubt be significantly less than that.
Even taking that upper distance of 10 lightyears between systems, journeying that far in under 10 millennia would require only exceeding mean travel velocity of 0.1% of lightspeed (or approximately 675,000mph). I do not see this remaining an insurmountably high speed for interstellar vehicles for very long. Assume that at least a couple of centuries are required at the destination star system to set up the colony and build the new colony ships out of local resources, similarly stocking them with a generational crew and anything else required for a generational colony ship, and then departing for the next star systems.
Finally, look at how far humanity has come in the last 1 million years; as long as we haven't been wiped out by cosmic misfortune or the worst elements of our own species? 1 million years should be more than enough time to colonise the entire galaxy!
Challenge Gregory SW Cooper to a bet on this prediction!