I suspected the Heller case had the most amicus briefs ever. I was wrong. Amicus Briefs Are Ammo for Supreme Court Gun Case:

The 67 amicus briefs in what is simply known as “the D.C. gun case” fail to topple the record number filed in the University of Michigan affirmative action cases in 2003 — 107 — but they easily fit within the top 10 filings at the high court.