Is there a way to increase the maximum horizon render distance in Minecraft?

Another way you can edit your render distance without tampering with class files yourself is to download a mod that does it for you. This mod has a 2x render distance and a 1.5x render distance version. Additionally, it offers the Better Cloud Mod.

Here is an image the author of the mod posted while using his mod:

Yup, you can edit your class files (an advanced mod, for sure) and make the distance beyond 512 blocks, the current value for "far".

...and there's a tutorial on world of minecraft for doing just that.

Since this question has been asked and answered, the state of Minecraft mods has moved on. The current gold standard in graphic improvement mods is Optifine, which is featureful, customisable, easy to install, regularly updated to address bugs and maintain compatibility with the latest version of Minecraft, and widely compatible with other mods.

Among its many features is an improved rendering engine which improves performance, and an improved draw distance that takes advantage of the rendering cycles that are saved by the improved renderer. The highest setting is now Extreme, which doubles the draw distance of the original Far, and it can be adjusted even more, in 1-chunk increments, so that you can find the perfect balance of distant visibility and performance.