I don't know why this got down-voted. The milky way is 100K light years across. It's 13.5 billion years old. It's entirely possible to blanket the whole galaxy with probes, without even getting into self replicating probes.
You have to make an assumption about speed - it certainly won't be more than a tiny fraction light speed. Eg voyager appear to travel now at less than 0.00006 c (16 km/s). Than turns 100k light years into ballpark 150 M year journey (no it doesn't its 1.8 bn years as mentioned in reply) . Still possible to cross 10.000 times - but that would be a straight line. I suspect the distances will grow somewhat fractally if you want to swing by any stars? (ed: so, not really very likely at all)
To imagine that voyager technology will define the maximum velocity of interstellar travel, a thousand years from now, ten thousand, a million, etc is short sighted. Maybe we never can exceed light speed, but we could get up to a reasonable percentage of it.
Sure, but it puts things into perspective. If we assume 1% of c is feasible - it's still a 10 Million year journey to cross 100k light years. That's much better - it gives 1 000 chances in 10 bn years.
There's current projects on earth seriously working on getting probes up to 1% of the speed of light. Just a small/light weight probe, a solar sail, and a large (but feasible) array of lasers.
It can't (unless you mistype into your plain android calculator app). Ed: Fwiw there was a typo there, too - it was supposed to say 150 million - but it's more like 1.5 billion.