I haven't had many accidents (thankfully) but I learned a lesson from one I had a few years back. That lesson was to always get a police report - and pictures if possible.
I was driving along the highway when the car in the lane beside me and slightly ahead cut into my lane without signaling or looking and hit the front side of my car near the driver's side wheel. We pulled off at the next exit and exchanged information. The young woman who hit me was 16 and driving her mom's car. She admitted fault to me at the scene and apologized (she was quite sweet, actually). She also had a school chum with her, the same age (illegal, I later realized; she could have had her license suspended), during school hours (probably skipping school). There was some damage to my car (and hers) but to me, it didn't look too bad.
Naively, my main concern was that everyone was ok. I never thought to call the police (or take pictures). I had to be somewhere so we left the scene.
Needless to say, my repairs were somewhere in the $3000 range and took three weeks. Although I always kept car rental on my insurance, somehow when I went to use it, it wasn't there.
Meanwhile, when my insurance company contacted the other party, the girl's mom began denying all kinds of things. I could barely believe my ears. She denied that the girl was driving or that she had a friend in the car. She also denied fault in any way. And I had no proof of anything because I didn't have a police report or pictures. It became my word against hers. Her insurance company should have paid for my rental if she was at fault, but they wouldn't because they denied fault.

(So ~ $40/day times 21 days = roughly $900 out of my own pocket.)
Fortunately, my insurance company was really good about it. They believed me based on the damage to my car (they have people that can reconstruct what actually happened even when people are lying). They said this type of thing happens all the time.

They had someone go to the other party's house to take pictures of her car, which the other party claimed had no damage (and I knew they did - in white, from my car, in the exact location that substantiated my story).

Last I heard from my guy was that they were threatening taking it to arbitration. I wound up fighting her insurance company who finally agreed to pay for my rental completely, but it was a PIA getting there (at one point they'd offered me a consolation prize of paying half, since I was "50% at fault". Ya, right).
If I had it to do over again I would have called the police to the scene. I'd also take pictures (I keep a camera in my purse for hobby's sake, but a camera phone work just as well). This happened on a busy suburban highway. I have heard people say they've sometimes had trouble getting the police to come to a fender bender if they're busy. I don't know how true it is because as I've said, it's only happened this once.
we did decide to call the police because my lower back was hurting within minutes of it happening.
And sometimes it's not until the next day that pain sets in.