I'm one of the founders of Crowdbooster, a YC S10 company built on Twitter.
Twitter, and specifically the API team, has worked hard to come up with guidance for where they think there are opportunities in the ecosystem.
They have very clearly told third-party developers to stop making Twitter clients [1]. Twitter wants to own the display of tweets since they are focused on making advertising their core business.
But there are definitely opportunities if you think beyond just making clients. The five main categories that Twitter specifically mentions are: Analytics, Curation, Content, Publishing, and Enterprise [2].
To echo PG, there are definitely still big companies to be built on top of Twitter. You just have to think big to begin with.
[1] https://groups.google.com/forum/#!topic/twitter-development-...
[2] https://dev.twitter.com/blog/ecosystem-showcase