One especially horrendous option is to start up the great gulf trail from the parking lot on rte. 16. 6 miles of trackless snow slogging later you will be at the base of the Great Gulf headwall. I would suggest doing this option either late in the season (April) or early in the season (December). Either way, bring snowshoes, prepare for the worst and start VERY early.
The way I've done it, which may be more on less feasible depending on your viewpoint, was to climb up damnation gully in Huntington, descend the Great Gulf headwall and arrive at the base of the route. We made it to the top of the route by 10 am after starting at 4:30 am from pinkham via this route. We did this in April and the snow was perfect neve. To try this earlier in the winter would be impractical to say the least.
To return, you will have to either cross the shoulder of washington near ball crag and descend the lion's head route or return to the Great Gulf wilderness via the descent of the headwall which is a WI1 III route in it's own right. If you descend the headwall, you can weave around and keep it pretty tame, but again this is snow condition dependent.
The crux is right at the base of the route and consists of easy ice bulges for a pitch to a rock and ice belay on the right. It's about WI2/2+ later in the season but may be more difficult earlier in the winter.
Another 900' of easy snow climbing with rock and ice protection on the left takes you to a little mini cornice near the very summit of mt. Clay. Finish at will out the top of the gully to the summit of Mt. Clay.
Simulclimbing this route is almost essential unless you want to spend the night out above treeline, simply because the approach and descent are so long.
3-5 screws for the ice sections and an assortment of rock gear. Pickets are helpful for the upper snow climb when the snow is in the right condition.
Typical Mt. Washington outerwear and gear should be obviously required. Remember, you are far, far, far from rescue so self rescue abilities are not a nice thing to have, but a neccessity.
You could get by with a mountaineering axe and it is probably more appropriate for the upper gully anyway, but two water ice tools will work.
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