A trip to the county courthouse (or the county GIS site) would probably reveal that your shared driveway is on a prescriptive easement, so adverse possession would not apply.
Actually, such clarity is probably not available. The deed to my property (and presumably also the deed to the other property that uses the driveway) contains a provision that “the alley” must be kept passable — but is not clear on exactly what constitutes “the alley”. Could very reasonably be interpreted to include the perpendicular “lane” at the top of the driveway, part of which is entirely on my property and part of which is entirely on the third lot, and none of which is on the property of the leaning fence lot. We (I and the third property owner) routinely have vehicles backing into each other’s portion of that lane when exiting our parking areas to go down the driveway.
This deed provision clearly dates back to horse-and-buggy era (both properties originally had “parking stalls” for horses at one side of this lane, long since replaced by open parking spaces) so the deed clearly wasn’t referring to “passable” by low slung sports cars, i.e. perfectly flat and paved. I very much doubt there is any other easement recorded anywhere, and the deed provision is so vague as to what land is being discussed that there might well be an adverse possession claim available, especially if the portion of the driveway that has had no use or utility to its technical owner for a century or more should become unpassable from lack of maintenance. Whoever pays to have it made passable would certainly have a strong claim against THAT property owner (but no claim to stop the third property owner from using the driveway).
Complicated enough yet? In all likelihood, that already happened many decades ago. I can guarantee that the driveway was neither originally nor most recently paved with any participation from the owner of the other half, and from what I know of the history my property and the third property (the one that owns none of the driveway), the paving was almost certainly originally and most recently done exclusively by an owner of THAT property pre-dating the current owner and the previous owner. That third property owner might well have a stronger adverse possession claim on the strip of driveway than I do (though perhaps not on the adjacent strip of unpaved ground), given that the only reason the driveway is still passable as required by at least two deeds, is that previous owners of the third property paid to make it so.
Frankly, the best resolution is for me and property owner number three (a good guy, who’s into small-time real estate investing) to simply buy out the leaning-fence property, and get the lots re-surveyed and re-drawn, and build up the raised driveway area to give each of us full ownership of a whole driveway accessing our respective properties, and then re-sell the lot minus the half driveway that no owner of the lot would ever have any use for (i.e. we’d end up paying nothing at all for the extra land). If and when it gets put on the market, I suspect that’s exactly what we’ll do. If any conflicts arise, we may pressure the owner to sell to us, under threat of legally forcing him to spend lots of money to shore up land under a driveway he has no use for.