Obviously not now, the road is closed and the temp is -6 degrees with 80+ mph winds. But for $20 you can drive yourself. I recommend having AWD. Here is the link to the page, could be a fun day if anyone around is interested this coming summer. I found this pic on the website too which peaked my interest http://www.mountwashingtonautoroad.com/
The last time I climbed Mt. Washington was around 1997. I was driving a chevy corsica and remember losing a hubcap on the way down. The brakes got so hot, both hubcaps on the front wheels warped, but I only lost one of them. I would up for a day trip out there this summer. It's a cool place with outstanding views.