Mount Soggy is a phenomenal snow climb with some of the best views around. In addition to snow, expect significant elevation gain (~6500 feet) from base to summit and a bit of glacier travel. It's a jewel and is located near a few other 7,000 foot summits such as Icicle Peak and Mt. Yukla.
To get to the base of Mt. Soggy, drive to the Eagle River Nature Center. From there, hike 9-10 miles up Eagle River Valley along the Historic Iditarod Trail. After passing through a canyon with 3000+ foot rock walls on both sides look on the left for a valley and waterfalls. This is the Twin Falls drainage. When you reach a small creek on the far north side of the Twin Falls drainage, walk up 100 feet and look to the right in the bushes for a hiker's/climber's/hunter's trail.
When you get on this path, follow it up 3000 feet or so to the hanging valley above. It's a beautiful valley and is a great spot to camp.
Also, you could approach Mt. Soggy from the Icicle Creek drainage. To do so, leave the Historic Iditarod Trail a little beyond 5 miles and head up a faint trail. Follow this trail up and above the cliffs and valley to the Icicle Glacier. It is recommended that you camp in the valley as the Icicle Glacier itself is subject to falling ice, snow, and rock. Note that if you reach the Icicle Creek, you have gone too far for this approach. The easiest way into the upper Icicle Creek valley is to the west (climber's left) of the cliffs that make up lower Icicle Creek.
Parking at the Eagle River Nature Center costs $5/day and $10 for 3 nights.
There are plenty of good campsites along Eagle River. The best for Mt. Soggy is a nice gravel bar spot just past Twin Falls. Also, the camping up in the hanging valley above is excellent with running water and spectacular views. Also, there are good campsites along the river at Icicle Creek and just before.
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SW Ridge From the Twin Falls drainage, go up valley until you see a deep groove between Mt. Yukla and Mt. Soggy up and left (climber's left). Go up steep scree, talus and snow to reach the Twincicle Glacier. Since there are some big crevasses on the Twincicle Glacier, rope up at the base and climb up and to the left up to Twincicle Pass. After this turn right and follow Mt. Soggy's SW ridgeline which snakes up and over a false summit before making the final snow climb. The ridge climbing is superbly fun with a few sections of Class 3 scrambling on the false peak. Otherwise it's snow climbing to 35 degrees or so.
North Ridge If climb Mt. Soggy from Icicle Peak, follow Icicle's South Ridge downclimbing a class 4 chimney to get down to a col between the two peaks. Then climb up steep 50 degree plus snow to Soggy's summit. West Face From the Icicle Glacier, you could also go up Soggy's West Face but beware of crevasses and avalanche danger. The West Face makes a snow climb all the way to the summit up steep snow to 50 degrees plus.