Was there a Jack in the Box on the west side of Fairfax, a block or two north of Johnie's and the May Company, in the early 1970s?
In my first year or so in L.A. I might have tried a cheeseburger at that or another location, and wasn't much impressed. I more or less thought of it as a third-rate chain until the early or mid-1980s. I'd moved to Redondo Beach to work in our office in Torrance, and sometime in that period they were advertising their then-new Breakfast Jacks. I tried one and was blown away by it. I couldn't get enough of those.
At that time the deluxe one was called the Crescent Supreme (they didn't say "croissant" then), with egg, ham, bacon, and I believe three kinds of cheese. It was to die for. It was such a favorite that when I'd been without Jack in the Box while living in Chicago later, the first thing I did on returning to L.A. in late 1986 was to get one of those back in my mouth. It was still to die for.
In all of those years I was only ever at two locations -- Hawthorne Blvd. near PCH in Torrance, and the one in the "Brown Derby" strip mall on Wilshire in the late 80s. And maybe the one near the museum if I'm remembering that right.