What is Ownership?
To have any influence over a planet you must be able to respond to it during the lifetimes of the people who live there, or you will never really own it in any meaningful way. Some people have made the argument before than you can not really control a place with more than a few weeks or months of signal delay just because such places require such autonomous governments to function, but for sake of what your overlord(s) might themselves see as ownership, I will use the loser definition of them simply having enough influence to tell a planet's inhabitants what to do.
Your rulers are immortal, but there is no way they can share this immortality and expect to remain in power; so, I will say that they probably rule over a population of human like mortals for sake of this answer. Since most human leaders can not control a nation for more than 30 years between when they are mature enough to take power and when they are too old to maintain it, I will say that anything that you can not have a 2-way communication with over the course of these 30 years can not be owned because no leader can be reliably communicated with or held responsible for their actions.
Feel free to adjust this window to suit your setting, but I will use 30 years as an example in my following calculations.
Limiting their empire by the Speed of Light
The larger your empire becomes, the longer communication takes between any two points in space. The Milky Way galaxy is just over 100,000ly across meaning that it would take over 200,000 years for a 2-way communication to happen between opposite edges of a Galactic Civilization. Civilizations, and perhaps entire species of intelligent beings will rise and fall in the time it takes the immortal emperor to even get word that they have decided to disobey him. So, following the 30 year limit, this means your immortal's initial maxim empire size will only be any star within a radius of ~15 ly. If your immortals started off on an Earth like world, that would give them a maximum realm of about 60 stars to rule over.
Limiting FTL by Inertia and Gravitational Sheer
Eventually your immortal beings will figure out some way to beat the speed of light, but faster than Light is not the same as instant travel. Let's say you use something akin to an Alcubierre Drive, in this scenario, you can now accelerate something to superluminal speeds without violating causality, but that acceleration takes time. Alcubierre Drives create gravitational sheer and inertia on the vessel they are accelerating which means you can only accelerate an Alcubierre Engine so fast before material science and the lives of those aboard hit their limits. In this scenario, ships with people on them are limited to acceleration speeds of about 1G (9.8 m/s^2), but smaller unmanned probes can theoretically survive much greater levels of sheer and inertia. Without knowing the actual materials that go into making an Alcubierre Drive, we can not say exactly what this limit is, but let's say for the sake of argument that an Alcubierre Drive can reach accelerations of 100G (980 m/s^2) before hitting the limits of the materials it is made out of which would tear it apart. (Feel free to adjust this limit to suit the needs of your story)
In this scenario, your emperor can now receive word of a rebellion from anywhere in the galaxy using such an FTL probe, and then send orders to a nearby fleet to put down the rebellion. This means that 2-way communication will require a probe to speed up and slow down at 100G over a given range twice. Sticking with out 30 year signal delay limit, this means your emperor will be able to control anything within a radius of about 23,130 ly. This means that he could at the very most control about 1/4 of the Milky Way Galaxy or about 25-100 billion stars.
Limiting the Multiverse by the Speed of Light
Once your immortals control a sizable portion of this universe; they would of course want to start dominating parallel realities, but most multiverse models assume that a multiverse exists as the result of additional dimensions. That means that multiple universes could be represented by new dimensions of space which themselves may be governed by the constant of C. So, by assuming a multiverse, we can simply calculate the volume of their domain as a glome (a 4 dimensional hypersphere) to make assumptions of how many places in the multiverse they can reach by following Alcubierre Drive like physics through multiple realities. Since the equation for the volume of a Glome is (1/2)π^2R^4 and R = 23,130 we can deduce that the maximum volume of multiverse you immortals can effectively govern would be about 1.411e18 ly^4 compared to the 5.18e13 ly^3 they could control just in our own reality. If we assume these parallel universes give access to similar cross-sections of Milky Way like galaxies as our own reality, we can extrapolate that to the volume of your immortal's 4 dimensional domain could contain about about 3.5-14 quadrillion stars within the 30 year soft cap of the homeworld. At this point, your "home world" would probably have to be some manner of dyson megastructure just to contain all of the necessary administrative infrastructure it would take to govern such a realm.
Of course, this is all just based on my somewhat arbitrary choice of 100G Alcubierre Drives and 30 year reins of planetary governors, you could always adjust your FTL and life expectancies for a vastly greater or smaller domain to suit the needs of your story, but the soft cap of mortal-life spanes multiplied by signal delay would remain your main scientifically limiting factors.