The Penguin's Nest (1) - S2-E27
Trivia: The guest cameo in this episode is by Ted "Lurch" Cassidy, who appears at the window as Batman and Robin climb up the side of the building. Batman advises him to go back to playing his harpsichord - which has been regaling us with the theme from The Addams Family since the scene began.
Batman's Anniversary (1) - S2-E45
Trivia: Jim Lefebvre - who played one of the Riddler's henchmen - was the National League's rookie of the year in 1965 and later managed the Seattle Mariners.
Batman's Anniversary (1) - S2-E45
Trivia: This is the only two-episode story arc in which John Astin replaced Frank Gorshin as the Riddler. After settling a contract and salary dispute with the studio, Gorshin returned to play the Riddler one last time, in the show's last season.
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