If you thought that hitting "Update All" on the Play Store meant you have the most recent version of an Android app, you'd be wrong. There are plenty of ways to get faster rollouts or even experimental builds for the adventurous.
Join Beta or Alpha Tests
At Google I/O earlier this year, Google gave developers the ability to create beta or alpha versions of their apps. These are subject to the usual early release disclaimers (could be broken, will probably change before release), but if you want the newest stuff, some apps offer beta programs. To enter them, you start by joining either a Google+ community or Google Group, depending on how the developer sets it up. Here are a few popular apps that have beta programs:
Download Separate Beta Apps
Other apps don't use Google's new Groups system and instead just release beta versions of their apps directly to the Play Store. These don't require any special enrolment or invitations to try them out. In a few cases, the beta app may even be the only version available. Here are just a few apps that offer direct betas:
Find APKs For Staged Rollout Apps
Some of the time, there aren't beta apps available but you still get left behind by a staged rollout. This is another feature Google introduced to developers at I/O this year, and the company makes prolific use of it itself. What happens is that a new app might "come out" one day, but it will only rollout to everyone over the course of several days or a week. If you're eager to get the jump on the update though, you can frequently find APKs for apps experiencing staged rollouts on sites like Android Police. Keep in mind while you're looking for these that installing APKs can be a source of malware if you're not checking the validity of your source.
Follow Developers or Apps For Direct Distribution
Another alternative you can sometimes find for early release apps is straight from the developers themselves. For example, Dropbox has an "early releases" setting within the app that will get you access to whatever cool new stuff Dropbox experiments with before rolling it out to the general populace.
You can also sometimes find independent developers who will distribute APKs themselves. This can happen when a courteous developer puts the app up while still doing a staged rollout, or for initial releases of new apps.
The Play Store remains the best way to get your Android apps, and automatic updating can make sure you keep up with the curve when it comes to new features. However, there are plenty of places where you can find even quicker access or experimental versions without straying too far into unstable territory.