This exchange between Batman and Alfred in the Batmobile, where Batman has a handful of coins and the clue "Ask me no questions and I'll tell you no tales. It all makes sense when you add it up."
Batman: Penny... Penny... Cent... Red cent... Copper! It's made of copper! Alfred: And "copper" is another word for "policeman"! Batman: And no tails means heads. Police... Head... Quarters! [...] Batman: Four quarters and one penny equal 101 cents, so... Police headquarters, room 101!
The silliest thing about this? Batman was going to go back there eventually anyway.
Leads into the above one. Three computers crash around Gotham, displaying only a riddle on screen: "Where does a 500-pound gorilla sleep?" "What's worse than a millipede with flat feet?" "How do you fit 5 elephants into a compact car?" Train of logic: the Riddler doesn't usually use such commonly known riddles meaning the answers to the riddles are a red herring. The riddles themselves all contain numbers: 500, 1000, 5. Convert to roman numerals and get D,M,V... the Department of Motor Vehicles!!! Which is exactly where the Riddler's goons are.
Chess Motifs: The Riddler trapped Batman in a virtual reality simulator. At one point, Riddler had Batman chase him across a giant chessboard, only to have the squares slowly disappear beneath Batman's feet. The villain's riddle is "Move according to the rules or it's the end of the day." Robin deduced that Riddler was playing with Batman's nickname The Dark Knight, and that in order to avoid disappearing squares, Batman had to move like a knight ("according to the rules" of chess); two forward, one to the side, to avoid the "end of the day" or nightfall (knight fall). Batman completes the puzzle by confronting the opposing King to simulate a checkmate.
Robin: You're looking at the guy who solved the Baxter's box in 37 seconds. Of course this time, I don't have a hammer.
Door Roulette: The first obstacle Batman encounters. This being the Riddler, however, the doors were all labeled with clues as to what was behind them. The door labeled "crazy intent", you guessed it, had an oncoming locomotive.
Hoist by His Own Petard: The Riddler can't resist hanging around inside the crashing program to give out one last riddle to Batman. It's to his own misfortune.
Me's a Crowd: In the end, Batman faces off against the Riddler, who demonstrates his mastery of the domain by duplicating himself. Batman realizes he can do so as well, and the two get into a ridiculous arms race of clones. Eventually the Riddler's consciousness is spread too thin for him to maintain the simulation, and it collapses, with Riddler's mind still inside.
Moral Myopia: The Riddler wants revenge on Batman for forcing him to become an outlaw. When Batman points out that this is because the Riddler tried to murder his former boss, Nygma dismisses his previous crime as "a personal matter" and blames Batman for getting involved.
Shout-Out: The aforementioned "loco motive" trap involves a train emerging from a fireplace, a reference to Rene Magritte's painting Time Transfixed in which the same thing is depicted.
That Man Is Dead: The Riddler is trying to enforce this by destroying computer and paper records of his Edward Nygma identity.