I can't even remember what service it was (this was mid 90's) but I once got an auto-generated password string from a site I registered on (might have been my online banking).
I ended up using that short string as a base password for everything and have continued to do so even today. I did this by doing the same thing you suggest, taking a small chunk and devising your own system for encrypting it while leaving it easy to recall.
Yes, there are certain site with overly simplistic password rules. For those I simply use a generic password that I would use on other sites as well. Obviously, those are what I consider "throwaway" logins and I am not concerned at all about someone finding the password as that site/service has no long-term value to me.
The sites/services that really mean something (web hosting panels, email, SSH stuff, etc) will always work with my scheme because they will always have robust password rules. If they don't, I choose another provider. This allows me to always know a password for something even though I may have not used the service in many months. Something like &Google-!@#$(mystring)$#@! or &Facebook-!@#$(mystring)$#@! is a simple enough way to do it, but you could certainly get more inventive if you like.