Ey, guys. So I kind of made this challenge on my own while playing TS2 in Belladonna Cove. I hope you enjoy it, and be sure to ask any questions and I'll edit this to answer them.
A/N: This challenge has been updated because I playtested it recently and realized it was vague and unachievable. I've made updates to the backstory, getting started, and rules sections. I also HIGHLY recommend downloading the InTeenimater mod. I have it. You'll need a teen pregnancy method. Maternity clothes for teens are REQUIRED, which this mod comes with. You can still find InTeenimater on Sims File Share. I will comment the link when I get a chance, but I have ADHD so you might wanna google it.

The Backstory___
Option A: You are a soon to be single teen mother in Belladonna Cove. You are around 16 and are early in your pregnancy and have been kicked out by your parents. The father of the child left town with his parents to cut ties with you and the baby. You've been couch hopping for a few weeks, but finally scraped together enough cash to rent a trailer to raise your child in. You're still in school, and are struggling to keep up with grades. You have found a low paying job, but can barely pay bills.

Option B: You are a single mother/father around 18/19 living in the city of Belladonna Cove. You had a child while in high school and your parents kicked you out so you had to raise the child alone. You started out living with friends for a little while, but it was time to live on your own. You're child is about 2-3 years old, and you've been out of school for about 1-2 years. --If you are a guy, then the mother left the child with you and skipped town. If you are a girl, then the father just left you alone with this kid to deal with.--

Getting Started___
Option A:
Start off with one female teen sim.
Make sure they aren't dressed to the nines. You can't afford nice clothes right now.
Give them the fortune aspiration, customize their personality and turn ons/offs.
Create a random adult sim. You can't move a teen in by themselves (unless you have a mod for that, in which please comment it, I want it). You'll have to make them related, but what I did was I moved the adult out immediately and then deleted them so they don't exist. You can also create your sims parents if you'd like and make the relationship bad and move them out for realism. Up to you!
For the baby, please either use the InTeenimater mod or follow the cheats method below. --Do NOT use the WikiHow method. It simply does not work.--

Option B:
Start off with one teen sim --male or female- (While you are around 18/19, you want a teen for job purposes)
Make them have cheap looking clothes and a shabby hair style. Nothing glamorous. You're broke, right?
Give them the fortune aspiration. Personality can be however you want.
(Since you are going to have a teen sim living alone, you'll have to have one adult sim too. So, just make a random adult, no need to customise. Move in the family, and then move the adult out or follow the same things I mentioned in Option A.)
Now for the baby. After you move in, you'll need a method of getting a child for you teen. As mentioned before there's Option One: Download the InTeenimater mod. Option Two: use a bunch of cheats and such**. -- Again. Do NOT use the WikiHow tutorial on getting a pregnant teen. It does not work and you'll just be spending a half hour messing with cheats--

**How to create a teen parent with cheats:

1. Create two adult sims and move them in
2. Ctrl-Alt-C, boolprop testingcheatsenabled true. *you may want to enable testingcheats before moving in, or you'll have to go into neighbourhood view and then back into your family*
3. Shift+click on either sim, click spawn, click Tombstone of L and D.
4. Click 'have baby with...' and choose the other sim in the household. **You can stop here for story Option A, continue for Option B**
5. click the stone again and click 'speed up pregnancy' then just fast forward
6. once your sim has the child, shift+click, spawn, more, sim modder.
7. click on the baby doll that spawns and click set age for whoever you plan to play as and choose teen.
8. shift+click the baby and click Make selectable/unselectable until you can play as the baby.
9. click the sim modder while playing as the baby and click set age... toddler.
10. shift+click one of your sims again and click spawn, rodney's death tombstone.
11. Kill off the sim you don't want to be the single parent
12. delete all of the cheat spawns.

Home Lot___
You've moved into the trailer park in the impoverished part of Belladonna Cove. You probably want the cheapest trailer there, however I recommend the 2 bedroom one that is available so you can have a roommate to help with bills. So, rent it and buy a few cheap baby things. --The Trailer park is in the very obvious impoverished section of the city. It's called "Main Street Mobile Homes" and it's diagonal from the Sims 2 water tower. You'll see the green apartment shaped 'plumbob' hovering above it. You HAVE to live there.--

Job Things___
(This is why you need a teen sim for Option B) You are only allowed to have a part time job until your sim ages to an adult (If you have the InTeen mod, you turn 18 seven days into being a teenager and will automatically be promoted to full time. This is fine). You must start at 5000 simoleons.

Rules___
*Note: Any broken rules result in failing the challenge. Also, there are no points for this challenge. You can make some up if you like.**
1) After moving in to a trailer and buying baby furniture, you must use the family funds cheat to bring your money amount down to 5000.
2) You can only sell 3 paintings a week.
3) You cannot dig for treasure.
4) You can sing and recite poems in coffee shops.
5) No money cheats are allowed except for when you are setting your money at 5000
6) You are not allowed to use any other cheats period.
7) Once again, you have to live in a trailer.
8) You have to buy groceries at least once a week. --Can be as many groceries as you like--
9) For Option B only: Since you are a teen, but are supposed to be around 18-19, you have to cancel the go to school action. --You're likely going to grow up badly, but thats kinda a given even if you would have started as an adult--
10) You have to have 50000 simoleons before you can move to the rich area of Belladonna Cove
11) You cannot order groceries, you must buy them from ONLY the Swift n Thrift mini mart.
12) You cannot go to any community lots other than Central Park east, Center Drip Coffee Shop, Swift N Thrift Mini Mart, and Desprit Deals Clothing Store until you've saved 7500 simoleons.
13) You cannot buy clothing until you've saved 6500 simoleons.
14) Since you live close to the ones you are allowed to go to, you can only walk to community lots until you've earned 7500 simoleons, as well.
15) You aren't allowed to buy anything you don't need until after you've paid rent, bills, and bought groceries for the week.
16) You're allowed to have a roommate.
17) You can move out of the trailer once you've been promoted twice at work (You still cannot move to the uptown area of Belladonna Cove until you've reached 50000 simoleons).
18) You can only rent until you've reached 50,000 simoleons.
19) You can get married and all that.


Goals___
Basically, the challenge is over once you move to the rich area of Belladonna Cove and live in an expensive house or apartment. It's up to you whether you live there or not. --The rich area of belladonna cove is on top of the hill in the northern region of the city. It's pretty flashy, you can miss it.
If you have University, then the challenge isn't over unless your child goes to college, too. --Basically, you don't want them to follow in your footsteps, so you want them to do better--

How you know you failed___
You've failed if:
1) You broke any of the rules
2) You're child is an adult before you've made it to the rich area of Belladonna cove
3) You didn't send your child to college. --only for university players--
4) You've been evicted.

You should have a lot of fun with this challenge. I really love the challenges that involve being really poor and working your way up! I hope you do too :D

Coming back to this almost ten years later is wild.

Interesting, but I just can't bring myself to get inteen, it doesn't play nicely with other mods I have. I may start this by having a teen from one of my big (and dysfunctional) families. There are a few smart, nice teens; one could move out with a yecky mom and her baby, then yecky mom will die, leaving the teen to raise the baby - sort of like the 2teens with4 younger sibs (what are their names again?) I always have an old couple move in with that family to help out (yes, I have a big family mod).

That is why I use S2L teen woohoo mod, I have not run into any problems with it.
That is Sims2Living folder on "Paysites Must Be Destroyed" site.