Box Elder Peak is among the highest peaks in the Wasatch range, but seems to lack much popularity because it's wedged inbetween the Alpine ridge (Lone Peak, Pfeifferhorn) and Mount Timpanogos. Nevertheless, it is an excellent peak to climb.
The most popular way up the peak is by hiking up the Deer Creek trail (American Fork Canyon) to a saddle between Deer Creek and Dry Creek (taking the trail up Dry Creek is longer with more elevation gain). I attempted the peak in '98 but from the saddle I didn't find the route up to the peak. I returned in 2000 and finished the job. Other routes are possible on this large peak, but I have no first-hand knowledge of them.
From Deer Creek, it is about three miles and 2,750 feet to the saddle, as compared to 3,950 from Dry Creek. The hike to the saddle is fairly steep and gets a good deal of sun. Mount Timpanogos and it's famous "glacier" will be visible as you get up a ways. The trail meanders through a large boulderfield just before reaching the ridge. At the saddle a fine view opens to the north with Pfeifferhorn's obvious dome most prominent.
The meadow would make an ideal campsite as it's big, flat and filled with grass. A trail follows the ridge south leading towards Box Elder Peak, still 1,500 feet above you. Follow this ridge about as far as you can on the trail (not taking one that descends to the west). You'll have to find a track leading up the ridge. As mentioned earlier, on my 1998 attempt I couldn't find it, but in 2000 I just started hacking thru the trees until I found something resembling a trail.
As you climb higher up this steep ridge the trees thin out and soon the trail (which becomes more obvious the higher you go) turns to mostly rock. In 2000 I had the mis-fortune of doing this climb on one of the hottest days of the year, 105 degrees in the valley. I had plenty of liquid, but in temperatures that were probably in the 80's up there, it was very taxing and I stopped to rest often.
Fortunately the views are outstanding as you get higher. You'll have a great view into the cirque of Box Elder Peak and the pyramid of Lone Peak in the distance. Mountain goats are common here and I watched a large herd of them leave my ridge and head west above some serious cliff bands. The peak looked like it was a long ways off but I ended up getting there quite quickly. Being by itself, the view from the summit is excellent. To the northwest you look right down that cirque out to Lone Peak and the valley far below. Numerous peaks from the Cottonwood Canyon area are visible. To the south is the obvious Mount Timpanogos. The ridge leading south appears to be very climbable, and I believe a route of sorts exists here.
On this incredibly hot day I was the only person on the summit. I signed the log and rested for a good half an hour before heading down. The trip down was much faster than up, but the heat was wearing me down. Many times on the trip back to the trailhead I had to stop in shady spots and try to refresh myself. A few hikers were actually hiking to the saddle in the afternoon heat! Finally a stream crossing early on in the hike gave me a chance to drown myself in water. The trailhead is somewhat confusing to find, and the trail itself has a few branches, so consult the local guidebooks to avoid wasting time on the wrong trails.
This is an excellent peak that is very under-rated. While I wouldn't label it a "classic" like Timp or the Pfeifferhorn, or even Lone Peak, it's still certainly worth the struggle at least once. The total climb is about five miles each way with a climb of 4,300 feet from Deer Creek. Plan on seven to eight hours for the climb and bring a lot of sunblock.