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UPDATE: (also added in comments)
I've been using LastPass for almost 10 years now, so I 100% agree that password managers are the way to go to manage the hundreds of different logins that we all have now.
I should have probably clarified this originally, but this suggestion is really for those passwords you can't (or at least shouldn't) store in a password app, like the master password for the password app itself, your network login for work, or the password for your own personal computer. These should also be the passwords that you should probably be changing more frequently as well!
Take a line or two from one of your favorite songs and then use the first letter (or corresponding symbol and/or number) of each syllable.
So for example
"Always Look On The Bright Side Of Life" could be represented as
Awlotbsol @W1otbs0l aW10+b$01
etc etc
You may need to write it down at first, especially when you first create or change it and need to enter it twice. But after entering it a few times, all you need to do is remember the line of the song and you'll remember your password!
And for passwords you need to change frequently, just use the next line in the song as your next password!