Of course, you'd probably prefer that your neighbor shoulder all those expenses. He just may be thinking the same about you. So for conversation's sake, let's ponder the legalities. First off, do you know who originally built the wall and why? If it was the neighbor or the previous owner, that certainly implies ownership and responsibility, particularly if the neighbor made a modification to his driveway necessitating the wall. Sometimes, though, such walls are the product of a previous civil agreement between neighbors. Ideally, this would be noted on your deed, title document or survey. Other times, the responsibility for such a "party wall" lies in the title documents of a homeowners association or planned unit development if your neighborhood happens to fall under either structure.
I might add that if either of you has done something to impact the wall's structural integrity since it was first constructed, or if the roots from your trees are pushing into the wall, that would carry weight, too, in more ways than one.
One thing is for sure: It doesn't benefit either of you to keep playing this game of chicken for an accident-in-waiting. Once you've looked into the background of the structure, I suggest an open discussion with the neighbor, assuming you're on decent terms. Tell him you're aware the wall is on his property but that you would be willing to help rectify the situation by footing a percentage of the repair costs. You might even volunteer to line up bids on repairs or reconstruction of the wall if that will provide momentum.
If your neighbor can't or won't pony up to at least split the cost, then you might look into options that would force his hand, assuming you're not content to pay for it all yourself. First, though, talk with your property insurer to get the lowdown on its coverage and liability guidelines in such situations, given the constraints of your policy. They've seen such scenarios before. You might suggest to your neighbor to do the same with his insurer.
If he still doesn't budge, talk with city code enforcement on the sly to see if the compromised wall was properly permitted or if it's subject to any building ordinances regarding dangerous property conditions. Oftentimes, however, the city will punt in such situations and say it's a private matter that you'll need to resolve with your neighbor or through an attorney.
Meanwhile, that wall is becoming increasingly unstable. Don't let it be a divider with your neighbor. Heed this old quote that some attribute to Sir Isaac Newton: "We build too many walls and not enough bridges." Here's hoping Newton's law of gravity doesn't take its course before you resolve the situation! Good luck.