You can add a standard string to all passwords, for example "Ab1#", so that it satisfies like 99% of sites. You do get in situations where you have to know remember some more information, but this can mostly be written down.
"You do get in situations where you have to know remember some more information, but this can mostly be written down."
That boils down to "this idea won't work"; "writing down my passwords" is the exact failure case we're trying to get away from. And I don't have that problem with LastPass.
The name of the game isn't to try to make this idea work at all costs. The name of the game is to find the best way to manage passwords. It's important not to lose sight of that in the argument. (If you look around, you'll see this particular cognitive problem come up a lot in engineering... never lose sight of the overall goal no matter how far into the trees you go.)
Unfortunately, at least a dozen iterations of this idea have come to my attention, to say nothing of who-knows how many hundreds or thousands of implementations of this basic idea there are, and they haven't succeeded because they really don't work in the real world.
If I write it down, I will lose it. I do have some passwords that follow a specific pattern that I can remember easily for sites I might need to access outside of having KeePass on a machine (haven't put it on my phone).
I love this idea, I hate having to have a actual database to keep after and secure, and I wish it could be done in a way that wouldn't require that, but sites have too many variables for passwords that for me, literally, this wouldn't work on sites I use every day.