According to the wiki article: "A few years later three-ring binders became the standard in the United States"

As for the number of rings: first mover advantage and network effects.

Paper is produced with three rings. Teachers buy three ring hole punches. Soon binder manufacturers target the most common standard. All other standards die off.

I have seen and used four ring binders in America. But three ring was the phrase that jumped out in my head.

The mind boggles.

3 rings: provides redundancy over 2 ring

4 ring: additional stability and redundancy

... 26 ring? At that point, you almost have more hole than paper. Is there a reason?

(4 rings for durability, 2 for convenience.)

Two-ring binders clamped at the top of the sheet.

Three-ring binders clamped at the left side of the sheet.

Four-ring binders were plastic instead of metal, and were a geared mechanical slider. One brand was the Trapper Keeper. ("Trappers" being the folders that matched them.)

At some point, the plastic four-ring binders switched to three rings to match the holes pre-punched in the loose leaf paper. But you could still get paper with 2+4 and 3+4 holes pre-punched in the left side that would fit multiple binder configurations.