Every battle background is composed of two layers, each with 327 possible styles (including "blank"/zero). The layer styles can be interchanged, such that Layer 1: 50 and Layer 2: 300 is the same as Layer 1: 300 and Layer 2: 50. Thus, there are C(n,r) = 52,650 possible different background combinations. Obviously, this many don't exist in the game--the SNES's graphical capabilities only allow it to properly render 3,176 of these combinations, and of those, only 225 are ever used.
The data for each of the 327 styles is bundled within the SNES cartridge. Tiles are constructed from various memory addresses in Earthbound game data. Each background also has its own distortion effect.
For a list of every possible background, visit this link.
It's pretty damn hard to find out which battle backgrounds correspond to which enemies. You can review the list of enemies matched with known battle backgrounds to try and get your favorite on the Suggested Layers option. If you discover a new one, please, make a pull request and I'll add it to the list, giving you full credit!
Because it can.
If you happen to know Shigesato Itoi, tell him about this! I'm a huge fan of his work.
Around 2010, Mr. Accident of forum.starmen.net provided me with the original battle background generation code, which was a C# project that uses the battle backgrounds as Windows screensavers. He also provided me with help along the way, and as far as I'm aware, he discovered the math behind it all, including the entire Offset function calculation. This port would be nothing without him.
In 2016, @kdex rewrote much of my shitty JS into nice and clean ES2016 code, for which I am grateful. As of
dc877f33, the entirety of his modifications have been brought over here. He is better at this than me.