Semo La in Tibet at 5,565 is the highest surfaced road, used actively by travellers.

Semo La (Chinese: ?????; Standard Tibetan: ????????) is a mountain pass situated in Coqên County, Ngari Prefecture in the central part of Tibet and gives access to the Changtang region. It is found on the so-called Northern Route, north of Raka and south of Town of Coqên in Central Tibet. Travellers use this route as an alternative access route to western Tibet and Mount Kailash, especially when mud makes access by the more southern route difficult.

The road crossing the pass was once an old unsurfaced track travelled only by a weekly bus and trucks heading west to avoid the boggy parts of the south of the country. The construction of paved road through the pass, Tibet Provincial Road S206, was finished in late 2015.

Elevation
The height according to the Tibet Department of Transportation is 5,565 m (18,258 ft); however, the signage at the road rounded it up to 5,566 m (18,261 ft). In 2005, a Catalan cartographic expedition certified the height to be 5,565.1 m (18,258 ft).

At 5,565 metres (18,258 ft), Semo La may be the highest asphalted road in the world. Khardung La was once thought to be the world record holder at 5,602 metres (18,379 ft); in reality, according to modern surveys, it measures 5,359 m (17,582 ft), 243 m (797 ft) less than previously thought.

Lonely Planet have an itinerary for the Northern Ngari:

https://www.lonelyplanet.com/china/northern-ngari/in-location/activities/a/nar/0899ecfe-ad83-4dd5-b21a-cfc4624ce790/1323946

Atop the World: The Xinjiang–Tibet Hwy With at least two passes above 5400m, the Xinjiang–Tibet Hwy is the highest road in the world. Approximately 1350km from Kashgar to Ali, this is an epic journey that can form a wild extension to a trip along the Karakoram Hwy

You'll need to do this as part of a pre-arranged tour, so if you're looking for a self-driven independent drive, you'll need to look outside of Tibet.

See also the Wiki for Extreme points on earth for other interesting places to visit!

The Tanggula Pass of China National Highway 109 (G109), the main road between Tibet and the rest of the China, reaches an elevation of 5,231 meters at its highest point.

The highway is fully paved and maintained, so it's passable in a regular 2WD vehicle, although winter conditions can be extreme. It's also the only road entry point normally open to foreigners coming to Tibet, although general Tibet restrictions do apply, which means you will need to have a licensed guide accompany you (you can still drive by yourself).

Not the highest, but perhaps much more doable than the above: Pikes Peak Highway near Colorado Springs.

Summit is 14115 feet (4302 metres). If you are legally allowed to enter and drive in the U.S. you can easily do this in a rental car. Try for July when it's open to the summit. When I went up in May 1995 it was open only to the 12000 foot (3650 meters) level. The air was noticeably thinner at that level.

Currently the highest answer on this page,

Abra Oquepuño, 4.873m

Some while ago, I did a bit of internet research on Macusani, a town in Peru. It sits at an elevation of 4,315 m (14,157 ft). It is on the Carretera Interoceanica, a relatively new road constructed between Brazil and Peru which was completed in 2011. Its highest point is at Oquepuño Pass, about 20km southeast of Macusani. It is described on the DangerousRoads site as follows:

Abra Oquepuño is a high mountain pass at an elevation of 4.873m (15,977ft) above the sea level, located in the province Carabaya in the region Puno, in southern Peru. It’s said to be the second highest asphalted road in South America. The pass is traversed by the Carretera Interoceanica, also known as Ruta nacional PE-34B. It’s an asphalted road.

I have done a virtual drive on highway 34B around Macusani and the road seems to be in excellent condition suitable for ordinary 2WD vehicles. I have never driven at that kind of elevation so I do not know how well a normal rental car might perform with the little oxygen available. Weather in general and sudden weather changes in particular are likely a significant hazard; I don't know the best time to travel the Carretera Interoceanica.

There is a timelapse video of someone driving on highway 34B from Azangaro to Macusani, dated 2015/05/06. They covered the distance of roughly 140km in a bit under two hours. At 23:45 into the video, they reach Abra Oquepuño. Road and weather conditions look excellent throughout the entire drive.

Update:

The world's highest motorable road passes through Umling La pass in Ladakh Union Territory of India. It is located at an altitude of over 19,300 feet and connects the Chisumle in the Koyul Lungpa valley and Demchok villages of Leh district of Ladakh.

Road to Khardung La pass (5359m AMSL) is the 2nd highest road in the world. The pass is a crossover between Leh, Nubra and Shyok valley. The Indian military vehicles drive through this pass to send supplies to Siachen post (the highest battlefield in the world). Any kind of vehicle can be driven on Leh-Khardung-La pass road.

ILP (Inner Line Permit) is required for Indian tourists and foreign nationals need Protected Area Permits (PAP) to drive in Khardungla, Nubra, Shyok region. One can apply for the permit on LAHDC Permit website.

Chacaltaya is a ski slope in Bolivia. The road to the base station might not be paved. The weather observatory is at about 17,000 feet (5200 meters). Outside of the Himalayas, this has to be one of the highest.