My first answer would be : 1 solar system.
While any race could potentially spread across an entire galaxy, jumping beyond your own galaxy are just TOO long distances to be plausible. There are plenty of stars within 10 light years that are totally at reach within normal science boundaries. And one could just start a colony there and launch from there beyond.. slowly spreading out over the whole galaxy.
But the long distanced between stars would make having a central center of government impractical, so it would heavy rely on local rules and local legions/fleets to govern those colonies, and send home the gains to the imperial home-planet.
With such long lines, I expect every planet to declare independence from the empire, and the empire being unable to do anything about it. As such a race would be spread out in the whole galaxy../ but it would not be 1 empire, but rather hundreds of thousands of them each comprising of the planets within 1 solar-system. Some might be at war with each other, some may trade with each other... some may try to form some form of federation of independent empires.
So in short a SPACE empire could just never exist... every trip would be a single trip.
First you would need energy. Since the energy needs need to be reasonable... (zero point energy is out of the question as is warp travel) also you cannot surpass 5g, or your crew would be tomato soup.
I also presume that the ship is magnetically shielded, so neither hard radiation nor tiny space particles would be a problem (you bend them around you, as the impact of even 1 atom at 0.5c on your ship would be impossible to armor plate against).
So I would think you accelerated constantly at a maximum of 5g for 6 months, towards 0.5c (before the energy needs outstrip the usefulness of going faster)
than fly in 7.5 more years to the nearest star and than start decelerating for 6 months. This way you'd make the trip in 8 years.
*if you were to turn around and return you would have flown for 16 years, yet almost 19 would have passed on your home planet.
However even if you were to communicate with your home planet.. 4 years to get a reply is just TOO long to be of any use for any empire.
(the roman empire collapsed and it had 2 WEEKS communication lines)
and the roman empire also gave strong power to local leaders.. with local legions, and the imperial army being slow to respond if needed if regional leaders rebelled against Rome.
In this situation.. any news about any rebellion would take 4 years to arrive.. and responsive action would take 13 years to arrive. It would be pretty hard to keep order in that kind of a situation. And that's why I presume that no empire as such could exist.
But supposing that all colonies are loyal to the home world (they're not like humans that like to self govern, but LOVE to please their empire-king or something like that) that it comes down to travel time towards the center of the empire.
Are there any outside attacks? Then no space empire as such could exist as responding would just take too long.. you would have lost the battle decades before you could send a counterattack.
But suppose you had none of that either?
Than it bottles down to lifespan or energy. Every 1 light-years of distance, means 2.25 years of flight. It takes for humans at least 25 years to be properly trained, and they can reach 120 years (when we reach our biological optimum). If you were to swap them for new crew for the return flight.. this would give us close to 100 years of flight enough to cross 50 light-years (giving us the maximum size for a human space empire).
However biological immortality is theoretical possible.. and while it could take half a million year to fly across some galaxies even at 0.5c, if one does not age or the ship is generational, it is plausible.
But energy, even a ship that could continue flying, would need to refuel, at the very least to be able to maintain life-support and keep growing food.
(even biodomes need HEAT, as space is cold and perfect isolation does not exist) also the magnetic shielding and artificial gravity would consume energy)
there are 2 methods to acquire this extra fuel (A) bring it all along, meaning more fuel load, giving every less return on cargo.
*refuel at a colony every 4-10 light-years, jumping towards the home planet in steps (having the disadvantage of having to decelerate/accelerate more often, increasing travel times and fuel use).
But suppose they would just DROP the cargo at the second base, and return... that way bigger and bigger ships would move towards the capitol, with each only carrying their tax and that o the worlds that paid it to them, towards the capitol. Well THAT highly strange race could get a civilization to run as large as a galaxy.
But it would not be a galaxy that would wield large armies or could respond to you taking one of their worlds in any reasonable way, any more than just a lone planet could.