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Due to my personal negligence, I didn't realize my passport is expired. I'm a USA citizen hoping to travel from USA to Colombia within a week. Is it even worth trying to attempt to renew my passport with an expedited service? Most estimates seem to be 2-3 weeks for "door to door" service.

edit: I'm aware I cannot travel with an expired passport. Wondering if I should try to cancel/postpone my flight now because renewing within a week is impossible, or it may be possible and how would I go about this.

edit: Just got an appointment with travel.state.gov. Their earliest appointment is only two days before the flight I bought. Also going to try to contact rushmypassport.com or consider walking-in to SF passport agency.

my resolution: Ended up getting a passport within 24 hours at the SF passport agency. Luckily for me, I live in San Francisco. Also, absolutely no lines when I went to my appointment this morning. Was in and out within 15 minutes, and will need to pick it up tomorrow.

Sadly, you will not be able to travel with a passport that has already expired. You'll need to renew your passport first.

The good news is it should be possible to get a passport urgently, but you'll have to travel to one of the Passport Agency offices with all the required documents, including proof of your upcoming travel, and pay the expedited fee. You can try to make an appointment online or contact the National Passport Information Center at 877-487-2778 (you'd have to wait to tomorrow to talk to someone though unless you have a life or death emergency, since today is a holiday). Since you need your passport immediately, the normal expedited process at a post office or other government office will not be sufficient. If you cannot get an appointment, it is usually possible to go to a passport agency as a walk-in, but you will be helped after those with appointments and can expect to wait a long time.

There is an important caveat to the accepted answer:
If you have an unexpired passport, you may be accepted on the flight but still be refused entry into your destination! Some countries require you to have six months valid passport upon entry, even if you're only staying say a week and have booked your outward travel.

I'm not sure how strict this is (a right to refuse doesn't mean automatic refusal), and when/where/how you can talk you way out of this with how much of a spare-validity-deficit.

Notice this is a caveat to the accepted answer, because a "next-day emergency passport" typically is valid only 6months (as opposed to the "expedited" ones that have normal validity); so if your destination country requires six months validity then your six-months-minus-a-week valid passport could still see you refused entry... [I have not checked this for Colombia.]

If you want an anecdote:
I have been in your situation, and the solution was what the accepted answer says. I hadn't noticed my passport had expired since two weeks. I had no reason to travel on a passport (within Schengen), I had a perfectly valid ID card that I always have with me as secondary ID (except that one time). I travelled UK->FR from my local airport, handluggage only, so I hadn't checked in beforehand and all was (erroneously!) passed as OK. I had my holiday, but was flying back on a Sunday from Holland. I had been trying to check in online on Friday, this kept failing and I didn't understand why (it didn't specify!). [If I had figured out the cause I'd have saved a good deal of money plus be home a day earlier.] I spent Saturday at a party and rocked up to the checkin desk, hungover, on Sunday, where the cause was revealed --- a now 3-weeks-expired passport.

Three solutions: In my own country I could get an emergency document, even on Sunday, in the main airport --- but taking train up&down there would make me miss the once-a-day flight anyway. Alternatively, I could take trains including the Eurostar to the UK, on the chance that the border control wouldn't spot the expired passport (like they didn't on leaving -- it did get scanned by them); but that was risky AND more expensive than accommodation plus flight next day. My embassy was nearby but closed of course, but on the phone I got the agreement that if I showed up on Monday at opening time (9:00), they'd squeeze me in first, and indeed by 9:30 I had my six-months-valid, hand-written emergency passport, in plenty time for my newly-bought flight.

Every single official from checkin-desk to border guard to boarding assistant did call their supervisor though, not believing my hand-written, differently-coloured passport to be actually valid. But my colleague who flies on a Liechtenstein passport has that even worse, as occasionally even the supervisors think that's a totally made-up country.