The Penguin's Nest (1) - S2-E27
Revealing mistake: When Bruce, Dick and Aunt Harriet get into what's supposed to be an elevator, there's solid concrete under the sliding doors where there should be a break separating the lift car from the building's floor. It's visible again in Part 2, when Batman and Robin use the same elevator. (00:04:30)
Answer: There wasn't one singular city used. It was shot in and around various locations in California, plus on a lot of sets and backlots. Ex. Some scenes were shot in LA, some in Pasadena, some in Santa Barbara, etc.
TedStixon