Is there any way to manually bring up the keyboard? In most cases where the keyboard is needed, it pops up automatically. However, I am trying to play Dungeon Crawl Stone Soup Webtiles on my tablet, and it does not bring up the keyboard. Is there any way to bring it up manually? I'm on a Nexus 7, running Jelly Bean. Ideally, there would be something that did not show up on the status bar, but sat in the notification drawer, and I could tap that to bring up the keyboard.

Actually, even better would be something that sat down in the bottom, next to the navigation controls, but I don't believe that's possible without modifying the framework.
As I see an alternative keyboard may solve your issue, and this seems to be an acceptable solution, and you even mention something you cannot find -- hereby I proudly present:

Hacker's Keyboard

Checking its Guide, there's in fact a section suggesting such a feature:


  You can configure the gesture used for this in the keyboard's "Gesture and key actions" settings menu. If the swipe action is too hard to trigger, try binding the "close keyboard" action to the "Volume down" hardware key. This won't interfere with normal use of the volume button while the keyboard is closed.


Though this applies to "close": If there are gestures you can configure, it might as well be able to have one for "open". The FAQ may reveal some more clues.

SaintWacko notes in the comments:


  To be able to open it anywhere, you go into the settings for the keyboard and check the box for 'permanent notification'. It will then keep an entry in the notifications which you can tap to bring up the keyboard at any point. 


So this would be an alternative possibility -- the hook being it is hooked to this special keyboard.