Imagine that less than a million years from now, probably in the hundreds of thousands, humanity has spread throughout the galaxy. There are afew aliens here and there, but that is insignificant, they are part of the same civilization. Anyways, for some reason our civilization has decided that the local group which will be bound to us forever doesn't have enough stars, so it decides, in less than a million years time, to send out a fleet of trillions, quadrillions of ships to take over other gravitationally bound galaxy groups and build abunch of megastructures to bring as much of the universe as possible into the gravitational neighborhood of the local group so that we will ensure our galaxy has as many gravitationally bound neighbors as possible to accompany it during the heat death of the universe. What are the specific reasons that humanity does this? unimportant. The question is how many galaxies could we bring together in an artificially created, gravitationally bound ultra-cluster centered on the local group? Assume that all fleets necessary for conquering these galaxies are sent out in less than 1 million years from now, that it takes each fleet around 1 million years to build the necessary structures for moving their particular galaxy and that the fleets travel to the galaxy at basically the speed of light (99.99%+). Assume no FTL capabilities for the return journey. What would be the fastest way to move these galaxies? Would we rely on less powerful methods like Shakardov thrusters to move near neighbors like the council of Giants to us and rely on more brute-force and expensive methods to get matter from more distant regions into our civilizational sphere? Yes, there will be other alien civilizations, some of which will resist human imperialism, but none of the others will be intergalactic. Humanity is trying to gather as many galactic acorns as possible to last itself as long as possible for the long winter that is the end of time and nobody else is going to be a significant enough power to stop them, though they may try.
It should be obvious, but this is not the same as asking "how far out can humanity spread" as my definition of "conquer" isn't that "humanity inhabits it" but that "humanity makes this galaxy/star/planet artificially gravitationally bound to the local group." The distance we could travel, assuming we travelled and light-speed, and then return to the local group is a good upper bound for how much of the universe we can conquer, but practical concerns of startup, movement, construction, and most of all the top-speed these galaxies could actually reach certainly makes the actual limit of conquest much nearer to us than the round-trip limit. I'm looking for answers that provide a conservative lower-bound estimate of what humanity could conquer and an upper bound estimate that requires really high effort and tight time-scales to actually achieve so that humanity's actual ability to succeed given the parameters I laid out would reliably fall somewhere in-between the two estimates.