The word that immediately comes to mind is "large". "Large category", etc.
Edit: Carl Mummert suggested this one, which I should have remembered myself and which is definitely widely used: "unbounded". "Unboundedly many Woodin cardinals", "unbounded rank", etc.
It seems to me that the simplest solution would be to say at the outset something like, "for us, 'large' will mean 'proper-class-many'..." or something similar.