I'd look for a 6" to 8" gravel base (such as ab3 which has some fines mixed in) for a sub grade. See if they run a sheepsfoot over the drive after grading it to compact it. Then sheepsfoot and roll the gravel to compact it. I forget the particulars, but there are different mixes of asphalt with one lift of a heavier asphalt and a top lift of a different type of mix which is what we used to put down for road construction. They should roll out each lift. A street minimum is 6" and usually 8" at intersections and then on up to 10-12" for semi-truck traffic. Have a wide turning radius at the street so you don't drive over the corners as is so common on street corners everywhere.
That said, you are probably going to be dealing with a contractor who does parking lots and driveways all the time and some will say 3-4" is enough. Many do parking lots that way putting asphalt on soil that has only been graded and the only compaction is from the grader or bobcat. That isn't enough. Go to a Walmart or other large shopping center parking lot and you will see what 3-4" of asphalt on bare ground looks like as there will be depressions where cars sit, etc.
Heavy soils like clay have lots of natural compaction when dry but really move around when wet and frozen which is why you need a compacted gravel base. Also make sure everything is bone dry before laying asphalt and that the temperature is above 55 degrees and the soil temp itself is above freezing or you will have future problems. When I was a paver and other heavy equipment operator for a road construction company, we used to go into town and do parking lots and driveways when the state and feds shut us down on the roads cause of temperature and wet weather. That is probably what they are doing to you.
It will cost a lot more but I would go with street construction standards. Also have them dig down 2 or 3" deeper for a few feet at the street entrance and maybe the garage as the transition area takes more of a beating.(you can see that at parking lots). Also remember that asphalt roads are designed to have seal coats put in every few years and check on the price for that as asphalt isn't maintenance free.
Most people are now finding that over the long run, concrete is better and cheaper and looks nicer, plays basketball nicer, and doesn't track in the house on a hot day so get a bid from both types of contractors. You may be pleasantly surprised. But some people will always look at the price up front and end up with a cheaper and less satisfactory job and then go whine about it later.
Finally, talk to your local city/county engineers and ask about local soil/road conditions and what they put down on their roads and parking lots in both asphalt and concrete and then ask to look at some since some of them even get budget conscious and don't put down enough. They should know the price/life comparisons. Good luck and post back with what you find out and what you finally do. Also some do a combination such as concrete where you park and play basketball, etc. and just do the drive portion in asphalt. My preference is concrete at 6" thick. Concrete also has subgrade compaction, weather, temperature, cutting expansion joints, thickness, and mix issues to watch for but a good job will last a long time.