Is there a way for me to rent a car in Los Angeles without having
existing auto liability insurance?

I don't see why you couldn't.  You could of course take the insurance
that they offer you to purchase.  They offer several types.  Have you
contacted companies such as Budget, Enterprise, Avis, etc?  I'm sure
you will find what you looking for if you just browse their website or
give them a call.

that's a lousy answer

Hey!!!

Thats a pretty lousy comment. The whole team of Google Researchers and
registered users (me) are here to help YOU, not the other way around.
You should be appreciative that exileonmain-ga acually took time out
to try and solve YOUR Query. Furthermore that is actually a good
suggestion that could save YOU YOUR money.

Dude, you don't think I already knew I could call them up or check their websites? 

The reason, though, that it was a lousy answer was "I don't see why you couldn't"
This person obviously doesn't have much experience with renting a car.
 A lot, maybe all (I don't know, that's why I'm asking) of car rental
places will not rent to you if you don't have existing auto liability
insurance.  That's the crux of the issue.  Yes I could have called up
every rental place in california and asked them but I was hoping that
someone would _just know_ the answer.  Not necessarily "this place
will do it"  but either "yes, some will" or "no, that's impossible."
So if you really don't know about what I'm asking about why respond
with [paraphrase] "duh, just google it?"  The guy didn't take ANY time
to try to solve my query because he didn't do anything.

Also, the team of Google Researches and registered users are NOT here
to help me, at least no more than I am here to help them.  We are
involved in voluntary economic exchanges which presumably we both
benefit from. maybe you should do a google answer on: "What is free
market capitalism?"