I hiked White Mountain last summer. It is not a wilderness experience but it was quite interesting in its own way, particularly, taking time to visit and hike the Bristlecone Pine reserve on the way there.
I managed to luck out and the day I picked was also the day that the Research Station was open for visitors allowing one to drive an additional mile or so and 1,000 feet or so further up.
The "trail" is actually a jeep road that leads all the way to the summit hut. The road up to the gate or the station is quite terrible, washboard, potholes, erosion runnels and sharp rocks and LONG. Did I mention L O N G! I noticed a few folks actually did carry extra spare tires. I did not get a flat but it is possible to gash the sidewall if not the tread.
If that doesn't let that scare you off - the LIGHTNING and hail storm at 14,000 feet might. It was absolutely one of the most terrifying events I've ever hiked in BUT survived as did the other tons of folks out there that day.
But it was also a wonderful vantage point, quite unlike the Sierra. Pretty views and you do earn a 14,000+ peak buut it is also desolate.
Camped at the gate - no formal anything there except an outhouse. Bring your food/stove/water and sleeping gear. I decided not to deal with a tent in the wind and slept uncomfortably in the car.
Hope I have not made it sound uninviting - it was yet another adventure and I'm glad that I did it.
Again, if you go, do NOT miss the Bristlecone Pines. ASTOUNDING. Somewhere on the board, I've posted a picture of White Mountain. In that thread there was a lot of discussion of the lightning storm.