We can wait a while to explore the Universe, but we shouldn’t wait too long. If the Universe continues its accelerating expansion as predicted by the usual model of cosmology, it will eventually expand by a factor of 2 every 12 billion years. So if we wait too long, we can’t ever reach a distant galaxy.
In fact, after 150 billion years, all galaxies outside our Local Group will become completely inaccessible, in principle by any form of transportation not faster than light!
For an explanation, read this:
• Toby Ord, The edges of our Universe.
This is where I got the table.
150 billion years sounds like a long time, but the smallest stars powered by fusion—the red dwarf stars, which are very plentiful—are expected to last much longer: about 10 trillion years! So, we can imagine a technologically advanced civilization that has managed to spread over the Local Group and live near red dwarf stars, which eventually regrets that it has waited too long to expand through more of the Universe.
The Local Group is a collection of roughly 50 nearby galaxies containing about 2 trillion stars, so there’s certainly plenty to do here. It’s held together by gravity, so it won’t get stretched out by the expansion of the Universe—not, at least, until its stars slowly “boil off” due to some randomly picking up high speeds. But will happen much, much later: more than 10 quintillion years, that is, 1019 years.
For more, see this article of mine: