I didn't necessarily want to rank my choices, so I thought about doing them alphabetically and then chronologically. I settled on the latter, and though I'm starting with a car from 1960, it's entirely possible that later on I may want to add one that's older.
I've alluded to my enthusiasm for 1960 Chryslers before, in this post about seeing a 1959 Chrysler 300E at the car show I attended last July and this one about a '61 Chrysler wagon on Curbside Classic. Those two cars end up as brackets for my favorite, the 1960 300F. The 300 "letter series" cars (link to wikipedia entry) were made from 1955-65; they were expensive cars with high-performance engines and luxurious appointments that never sold more than a few thousand units per year, but served as halo cars for the brand.
Initially available as a two-door hardtop coupe, a convertible was added for 1957 and continued through 1965, except there was no convertible offered for '63. Production for 1960 was just over 1200 cars total, and only 248 of those were convertibles. I have seen restored examples for sale for as much as $75,000, though I have no idea if any of them have sold for such prices. There is one 300F convertible currently for sale on the classic-car site Hemmings for $199,000!