One bucket list item I have is to take a back country flight there.
A clarifying question: are you looking for a flight OVER some beautiful backcountry (flightseeing), or a flight involving LANDING in the backcountry, or a backcountry-flying LESSON?
There are lots of operators that will do the first. Some specialize in things like bear viewing, or National Parks (Kenai Fjords / Denali / etc.). Just about every town (Girdwood, Talkeetna, Homer, Seward, etc.) will have flightseeing tour operators of some sort -- some fixed wing, some helicopter. From Anchorage, I would look for something departing out of Lake Hood -- Rust's, Regal, Trail Ridge, etc. -- just so you can say that you've experienced the world's busiest seaplane base.
Fewer operators will do the second, unless there is a reason to be landing in the backcountry, such as to get to a remote hunting or fishing site, or to drop you off for backpacking, or something like that. For instance, Wrangell Mountain Air dropped me and my family off in the middle of nowhere to go backpacking in Wrangell St. Elias for a few days. For hunting/fishing, the flight is often part of the package that you buy from the guiding operation. That being said, there are some "bush experience" tourism flights to be had, with a little bit of googling. The operators out of Talkeetna (K2 and Talkeetna Air Taxi are the two big operators there) have a "land on a glacier!" flight, which I've never done, but would scratch that "thrill-landing" itch. Alaska Air Service advertises a flight where they land on the Knik River. There are probably others.
Even fewer places do backcountry instruction -- insurance is no doubt prohibitive. Blue River Aviation in Palmer might do it -- they've got some Super Cubs. Alaska Floats & Skis up in Talkeetna has a "bush pilots course" which I know nothing about. Another option might be to take a floatplane lesson at
https://alaskafloatratings.com on Trail Lake (Kenai Peninsula), who can also package your flight with a cabin rental.
Flight aside, if you've only got four days, I would spend it exploring the Kenai Peninsula. It's close enough to explore by car, and gorgeous in pretty much every corner of it. Yes, there are cars and tourists -- not "real Alaska" as the locals will snark -- but there are lots of trailheads into the mountains, and landscapes and wildlife to be seen, and it'll whet your appetite for coming back in a Serious Way.
@FORANE is not kidding about that drive south along the Seward Highway (locals do not call it "Highway 1") -- it is one of the most beautiful drives in the US.
Other places in the state (like Denali Park, etc.) are also awesome, but are more of a driving commitment. Remember that everything in Alaska is farther away from everything else than it looks on the map.
In Anchorage: eat at:
F Street Station: a pilot's bar downtown
Spenard Roadhouse: bourbon enthusiasts with something called the "bacon of the month"
49th State Brewery: has sweeping views of the Cook Inlet
Drop me a line,
(I'll PM...)
