That is tough one. It really depends on what one considers a great cheeseburger and LA has so many excellent contenders. There are, of course, your chains which originated here and help define the southern Californian experience. I would put In-N-Out and Tommy's at the top of that list and I love them both. However, I just discovered and fell in love with The Counter. There are a few throughout LA (www.thecounterburger.com). I used to love The Munch Box in Chatsworth when I was a kid. I grew up there and would get a chili cheese hickory burger and root beer float after baseball games down the street. We used to walk the railroad tracks back before anyone cared about kids walking on railroad tracks. I also a pig fan of the pastrami burger at The Hat. I also have fond memories of The Apple Pan on Pico. I am sure there are so many other little whole in wall joints that just happen to pump out the best burgers in town. I still have to try the Cardiac Burger at The Bucket in Eagle Rock. It is legendary and I must conquer it one day. Obviously, I will have to live at the gym for the next week to correct the damage done and not "kick" The Bucket.
I am partial to In-n-Out but I found I really like Fatburger too. Next time I will have to try it with a fried egg. A good cheeseburger has to have a good ratio of meat to bun and it helps if it has more than that plasticy american cheese
I love Fatburger. I often get the Kingburger with the fried egg and bacon. Several years ago while working at a theatre in West LA we went to Fatburger and ordered a whole bunch of delicious and brought it back to the theatre for the crew. The fun part was that we were currently working on a local synagogue's high holy days and they were all fasting that day. We weren't though and we were hungry. We ran it through the building up to a break room but got so many mean stares when they smelled us and saw the bags we were rushing past. ;)
Don't know of any places downtown. Maybe some of t he steak places also serve a good hamburger. There is a place upstairs from the library that serves good tiny burgers during their happy hour. Remind me and we can check it out sometime.