You don't need FTL to explore the entire galaxy in 1B years. The Milky Way is only 100k ly across, so your fleet would only need an average speed of .1c to see

You don't need FTL to explore the entire galaxy in 1B years. The Milky Way is only 100k ly across, so your fleet would only need an average speed of .1c to see it all. E.g. a robot fleet with a program to "see all parts of the galaxy" would colonize, refuel and rebuild another fleet, and send it off to repeat, and easily reach the entire galaxy. (The tech we're missing is a way to turn solar power into anti-matter, then power your ships with slow trickle matter/anti-matter annihilation.)