Even our oldest running computers are young by human standards, and downright primitive compared to what we are producing today. Machines are not immortal, at best they are easier to repair than lifeforms but that's not of much use if there's no one to repair them. The voyager probes are in their 40s, and it's not looking like they're going to make it into their 80s. While we can certainly build more robust machines with more redundant components, the fact is without FTL the overwhelming majority of a galaxy is tens of thousands of years away at a minimum[0]. Building things to last over such timeframes in a harsh environment like space may not even be possible, nonetheless practically achievable.
Even if one assumes we can make such long lived machines, the question becomes why? By the time such a craft reaches its destination, every being that worked on its construction, and all of their descendants for many generations, and indeed possibly the civilization itself, will be gone. Even if some institution still remains, it will be too far away to communicate with, nonetheless transfer anything with mass. Whether operated by cloned beings or AI, this new settlement is in no way a part of the originating civilization, at least not any more than some ancient Egyptians shipped off 1000 lightyears away would nowadays be a part of ours. There may be legitimate reasons to set up a few colonies, such as backups in case of solar-system scale catastrophes, but by the time you've expanded out even just 100 lightyears (0.0001% of the galaxy), there are already tens of thousands of systems in your "empire". Of course civilizations don't need to act rationally all the time, even though they might derive no benefit they might do something just because they can. But how many times are they going to do this just because they can? 10 times, 100 times? 10,000 times? Eventually the novelty is going to wear off, and those massive quantities of resources being spent on these extremely expensive spacecraft are going to be repurposed for the betterment of the civilization.
Now let's say that, despite the extreme technical challenge of travelling more than a few tens of lightyears, and despite the complete lack of rational reason to do so, a civilization decides to just keep expanding anyways. Eventually they are going to encounter another civilization. This continuous growth is a threat to any other civilization - even if they agree to leave their neighbors alone, they are still ravenously consuming the resources in the area that such a civilization would want to have access to in the future. It is in their best interests to get the expansionist civilization to stop - possibly by diplomacy, possibly by other means. The expansionist civilization is squandering resources on its expansion, it's likely a young civilization that only recently developed the means to expand, its periphery is thousands of years less advanced than its core region and likely very underdeveloped, it has no means to move its assets to the periphery or concentrate its forces in the short term, and it has extremely long supply chains in the long run. Conversely the neighbors have been using their resources efficiently, they may have been around for a very long time, and they are operating close to their core region. The expansionist civilization will get in a fight eventually, and it will lose. Even if it doesn't get defeated so utterly that it is forced to stop expanding, eventually it will just encounter another civilization that will stop it from expanding.
The only scenario where continuous expansion is possible without FTL travel is if one of the very earliest civilizations in the universe has an inexplicable drive to expand and does so fast enough that no other civilization can arise fast enough to snuff out all possible competing civilizations in their infancy. That we have reached this point in our technological advancement long before a non-FTL civilization could expand throughout the visible universe strongly suggests such a scenario is impossible.
[0] Some people may feel inclined to bring up Von Neumann probes - craft which travel to other star systems, repilicate themselves several times over, and then launch more such probes to more solar systems, thus allowing for exponential growth. This technology doesn't actually make expansion any faster though. Building a probe at A, sending it to B, where it builds a second probe, and then sending that to C will always be slower than building two probes at A and sending one each to B and C directly. There is also no benefit to longevity - building a new probe is equivalent to replacing every component of the original.