Dropbox with KeepPass on Windows / KeePassX on Mac / MiniKeepPass on iOS works beautifully for me.
The biggest issue is that the passwords are usually copied via the clipboard, so any program accessing the clipboard could copy your passwords. It's of course up to your own assessment whether this is a problem worth consideration or not.
Unfortunately many places consider this information sensitive and protect it fiercely.
Note that "a good memory" doesn't "scale" or remain constant over time. Plus more importantly you are assuming that something minor or major won't happen to you where for some reason you aren't able to remember things very clearly like you thought you would be able to.
I don’t think you need to worry about that. See https://news.ycombinator.com/item?id=3140898
Generate your passwords with a simple algorithm with a servicename and a keyphrase. Works really well.
Comparing this to not using password managers, (in my opinion) I would say that LastPass is both easier and safer alternative than not using one.
But I have a good memory and I trust it much more than my ability to keep things organized and make routine backups. It's easy for me to generate more or less secure passwords that I can memorize, like:
shrebangodiKe24+ binarKedonado!3297 Miregofinar--0009
etc.
Those will last for a few more years, until password crackers will become so fast that it will be impossible to remember any kind of secure password anymore.
Then, I'll be screwed.