In California, the maximum security deposit allowed by law for an
unfurnished apartment is 2x monthly rent. Pet deposits, key deposits,
cleaning deposits, etc. are considered security deposits. Security
deposits can NOT be "non-refundable".

With that said, I have an apartment with a monthly rent of $1050. I
have paid a security deposit of $2100. The lease allows for a cat,
with an additional $200 pet deposit. Here's the question: can I still
bring a cat into the apartment, even though the owner is not allowed
to ask for an additional $200 deposit?

In other words, because I cannot be asked to pay the deposit, does
that mean that I cannot have a cat (since the deposit cannot be asked
for or paid)?

http://www.leginfo.ca.gov/calaw.html

1950.5. (c) A landlord may not demand or receive security, however
denominated, in an amount or value in excess of an amount equal to
two months' rent, in the case of unfurnished residential property,
and an amount equal to three months' rent, in the case of furnished
residential property, in addition to any rent for the first month
paid on or before initial occupancy.
   This subdivision does not prohibit an advance payment of not less
than six months' rent if the term of the lease is six months or
longer.


Check out the second paragraph.  The casual web sites don't mention
that little detail.  Six months rent is a lot of money, and seems to
lend much bargaining power to the landlord.

Here's a web site that you might like:
http://www.renterswoescentral.com/