I don't think you are going to find many, but some of the platformers, like the Abe's Oddyssey games, might work OK. Frankly, most of the games that do support gamepads already are designed around at least a multi-button setup, if not the full dual-analog with D-pad and multi buttons that is the (fairly) common standard today.

Well, most of the older DOS games with joystick support should work pretty well. Joysticks only had one or two buttons back then.
Anything (sold here) that doesn't use DOSBox though will almost definitely expect more buttons. And many of the ones that DO use DOSBox will still expect more buttons :p

I'd be a little concerned about the D-pad. Even if it is a DOS game that only expects a couple of buttons like older joysticks, it likely also expects analog direction control, not digital.

Many older joysticks were digital, no?

Mostly those that used Atari VCS/2600-style sticks, like the C64 and Amiga. They may have had a few analog sticks, but most of them were digital, with microswitches representing the directions rather than a pair of potentiometers. (Fine for most games, but I don't see flight sims being very bearable with a microswitch stick...)
PC joysticks were predominantly analog, though there may have been a few digital ones here and there. (Old ADB Mac sticks were probably mostly analog too, but I'm far from certain on that one.)
Oh, and to further add confusion, the analog/digital distinction applies to input devices in two ways-the direction controls AND the interface. 1990s joysticks started having a digital gameport interface (one that does NOT work with your typical cheap gameport-to-USB adapter, which only works with the old analog gameport interface), but with analog axes for the main joystick movement for precision's sake, of course.

It gets even more confusing when you try using a modern gamepad instead of a joystick with an older game. I've got a Logitech Dual Action and a Saitek P990, both similar to a Playstation-type controller. Most of my older DOS games are able to use at least one of the analog sticks and most of the buttons (usually the shoulder buttons are ignored), but some refuse to recognize the D-pad on either controller, others see it as a POV hat switch and refuse to see it as anything else. A lot of it depends on the game, but it can be a crapshoot trying to get some controllers to work natively. I usually end up resorting to a keymapper like Xpadder to get the functionality I need/want.

Another World, maybe...

Nope, Abe's Odyssey has quite a few action buttons.