It is normal. The asphalt never completely hardens. The weight of parked vehicles will, especially in hot weather, cause it to deform and to crack at the edges.
Keeping vehicles as far from the edge as possible and applying vegetation killer in any cracks should slow the effect.

As Randy427 said, it is normal. Take a walk through your
neighborhood and you will see. That used to happen to
my driveway, until I had curbing installed. Now the
asphalt has something to butt up against.
But curbing is not cheap.

Thanks for the answers. Can a grass that spreads underground actually poke through asphalt? It pops through areas not cracked.

Grass seed, and the seeds of many wild plants, can get into
those tiny cracks that you speak of. And then will sprout
and continue to grow. It is not necessarily grass that
is growning 'up' through the asphalt.

What about repairing cracks in asphalt? The cracks I have at the edges of the driveway have been patched with the crack filler tubs you can buy at HD but they always seems to spall up and reform the cracks. Would the cold patch stuff form a better bond? or is there a better product? It would take too much caulk so the tubes of driveway caulk won't be cost effective.

You will NEVER eliminate those cracks at the edges,
unless there is something there to hold it in, (like
curbing).
As they say in New York City: FUGETABOTIT !!!!!

I had the driveway seal coated today and it looks really good. Upon further inspection yesterday my concern about the spreading grass poking up through asphalt still holds true. My bluegrass is definitely workings its way up through the 3 inches of asphalt. Is asphalt really that soft that a spreading grass can burrow it's way through it?

yes

Thanks! I called the seal coat man and asked him the same thing and he said he sees it all the time with the spreading grasses. I just wanted to post that here in case someone has the same question in the future. He did tell me to kill what's popping through to avoid damage on the edges. I'm the king of Round Up so know problem there.