Big Four has big fame in the Northwest. At least that's what I've always thought. Its fame stems from two seeds of awe: the stupendous North Face and the ice caves at the foot of the North Face. The North Face is plainly visible from the highway (Mountain Loop Highway) at several locations and from the viewpoint/picnic area near the highway. This face is 4,000 ft high. It rises like a palisade from the ice caves in a series of high-angle ribs and gullies. The ribs rise up to form five distinct nubs that look like brass knuckles. Indeed the symbolism is apt, for there is no easy way to the top of this peak and it might beat you up if you try. The peak is everywhere rugged. The easiest route is from the east (from Dry Creek). The North Face relents with difficult ice climb routes but it is generally avoided in summer (loose rock). The NE Ridge contains a hard rock route called the Tower Route.
The ice caves have been a tourist trap for years. An inn was even built at where the present Big Four viewpoint is located. The caves have melted back quite a bit over the years but they are still worthy of a visit (especially since the approach trail is short). What are the ice caves, then? Essentially, there is an avalanche debris cone that forms (or has formed over the centuries) at the base of the North Face. This cone is known as "Rucker's Glacier." Waterfalls plummeting down the face enter at the back of the cone and subsequently flow under it.
The water eventually exits at the base of the cone in a series of streams. The combination of ground surface heating characteristics and air flow has excavated large tunnels under the base of the cone. To enter these tunnels is quite dangerous (people have been killed when caves collapse), but they are pretty interesting. Also, people often climb up on the snow cone. The higher you go the steeper it gets (ice axe mandatory and probably even crampons). Some climbers even practice ice climbing in the moat behind the cone.
The mountain does not take its name from the knuckles on its summit crest (there are actually five knuckles). Instead, it apparently got its name from the giant "4-shaped" snowpatch on the East Face. I've never seen this snowpatch but there are old photographs of it in existence. See Schroder's post here. The same "4" is also suggestive in this picture.
Rock of Big Four in its upper portions is mildly metamorphosed phyllitic argillites and slates of the Chilliwack Group. For the NE Ridge and lower down, rock is comprised of Swauk-Chuckanut sandstone and conglomerate. Many mining claims were placed on the mountain but none of these proved fructuous.
There is really only one approach to Big Four that is worthy of mention here. The intrepid explorer might approach the peak from the south (from Spada Lake) but, if he/she does, they won't be coming here for the information. In my case, I will only elaborate on the north side approaches (from Mountain Loop Highway).
To get to Big Four Mountain from Interstate 5, take the US-2 exit at Everett. Go three miles east on US-2 then go left up the hill on SR-204 to Lake Stevens. Go left (north) on SR-9 for two miles then right (east) on SR-92. This road later "becomes" Mountain Loop Highway. In about 8 miles, the highway arrives at Granite Falls. Alternatively, from Interstate 5 in Marysville, you can take SR-528 east for four miles to SR-9. Turn left (north) on SR-9 and go two miles to 84th Street NE. Go east on 84th for five miles to where it junctions with SR-92 a few miles west of Granite Falls. From the major intersection in Granite Falls, drive the Mountain Loop Highway east for 25.5 miles to the Big Four viewpoint and ice caves trail.
If coming from Darrington, you can take the Mountain Loop Highway south from there. Go over Barlow Pass then continue west downvalley (South Fork Stillaguamish River) five miles to the Big Four viewpoint.
The North Face, Northwest Ridge, and Tower Route climbs generally begin from the viewpoint. Take the one-mile trail to the ice caves then go whichever direction is required to get to the start of your route. The trail sees a lot of traffic and generally utilizes boardwalks to cross the boggier sections.
There really aren't any permit issues for climbing this mountain. There may be a Trail Park Pass issue, but hopefully that fatuous requirement will go the way of the dinosaur soon. For the Dry Creek Route, you park at the side of the road at milepost 27. There is no "trailhead" there.
As mentioned, there are winter ice climbs on the North Face, so really this mountain can be climbed year-round. The viewpoint is at 1,700 ft. The Dry Creek Route starts from the road at 1,800 ft. As long as you could get to that point, you could conceivably climb this peak.
There are several campgrounds on Mountain Loop Highway. Some of these are as follows (this is not an exhaustive list):
Turlo and Verlot -- 11 miles east of Granite Falls, 14.5 miles west of Big Four viewpoint
Gold Basin -- 13.5 miles east of Granite Falls, 12 miles west of Big Four viewpoint
Wiley Creek -- 15 miles east of Granite Falls, 10.5 miles west of Big Four viewpoint
Coal Creek Bar -- 23.3 miles east of Granite Falls, 2.2 miles west of Big Four viewpoint
Beaver Bar -- 24.5 miles east of Granite Falls, 1 mile west of Big Four viewpoint
There are plenty of places to simply pull out from the highway and camp (usually at the river bank or at a tributary stream bank). You will see people doing this all along the road. Camping is not allowed at the viewpoint.
In terms of camping on the mountain, I will defer comment. This mountain is a day-climb...unless it isn't. In the case that it isn't, you'd have bivy gear with you. Right? Since the peak is so rugged, your impromptu camp will be at whatever flat spot you can find. The Dry Creek basin on Big Four's east side is fairly flat until you start going up.