I couldn't leave out Town Hall! It's my favourite building in Adelaide, which is why I also volunteer there as a tour guide!
Built in 1866, it is open to the public during business hours and most of the areas are free to wander around- including the balcony which has been host to a number of important/famous people. Most notable were the Beatles in 1963, who were greeted by over 300,000 passionate fans on the street below. That was over half the population of Greater Adelaide at the time and the biggest turn out the fab four every received.
Tours are free most Mondays at 10am and run for an hour, and hey... we might run into each other!
128 King William St, Adelaide SA 5000, Australia