The most often climbed mountain in Alaska. (Technically the name is "Flattop Mountain" but nobody calls it that.1) The mountain lies within Chugach State Park, southeast of the Anchorage Bowl, at the end of a 1.5-mile trail, and is 3500ft/1070m high.
From the Glen Alps trailhead on the Anchorage Hillside, the actual altitude gain is only around 1300 feet, despite the mountain's height. The trail is an easy hike until you get close to the top of the mountain; then it gets difficult. This harder portion has several steep rock climbs and is somewhat dangerous; there have been deaths.
The general consensus, however, is that once you get to the top, it's well worth it. The views are nothing short of astounding, with the Chugach Mountains to one side, and the city and Cook Inlet below to the other, as well as, er, more mountains. If you're lucky, there won't be too many other people hogging the good view spots. ;)
If you're a tourist visiting Anchorage, spend your money as fast as possible and get right back on that damn airplane expect to hear about Flattop fairly often. This is the place everyone has either gone or is planning to go.
A couple caveats:
- Crowds: I did say it was the most-climbed mountain in Alaska, didn't I? Flattop can be a mecca for anyone in the Anchorage area who wants to get outside a bit. Yes, including the tourists. (This can become an issue on the more difficult trail sections, as if you're not careful you can have rocks kicked down on you by those above.) This goes all day, by the way. Flattop is an excellent place to go really late (in summer, one has to) to watch the sunset.
- Wildlife: There are two kinds you may or may not run into. Bears you can ward off by just being sure to make plenty of noise. "Bear bells" are handy for this purpose. Moose... well, all I can say is don't try to pet them, and don't get between any mothers and their calves. Unlike the above two, you will experience mosquitoes, and you're a fool if you don't bring some kind of protection.
Getting there
Taking O'Malley Road east, turn right (south) on Hillside Drive. Turn left (east) on Upper Huffman Road and continue for about half a mile until the road stops and forks. Take a hard right. (By the time you're even as far as Hillside Drive, you'll be a couple thousand feet up, and will definitely be feeling the altitude a bit, unless you're from Denver. This is normal.) Follow the road to the trailhead.2
Notes:
1. And anyone who asks why it's called "Flattop" shall be shot without trial.
2. Note to would-be smart-alecks: A quick glance at any map implies you can take Huffman Road straight out to Toilsome Hill without using the longer directions I gave; the maps lie like a mofo. If you try to do this, you will find Huffman turns north a mile short, becoming Birch Road, and you get dumped right back on O'Malley anyway.
3. Except in July.
Admit it, you looked for that missing 3 superscript, didn't you? | 
