A big Meta Awesome to Seiji Takaiwa and Makoto Itō, both of whom have done the suit work for Double and manage to convey with their body language that there's two people making up this rider. Check out the series Crowning Moment of Funny page for a few stand outs where conveying there are two minds in there sells the joke, and there's a few examples here too.
Double and Dango's dance-battle session at the "Find the C" arc climax; they were matching overwhelming a super-speed Dopant, which previously snatched four of Double's Gaia Memories like candy from a kid, all the way without a pause. The dance music only added more points to the scene.
Doubles as a Funny moment when, as it begins, Shotaro is flabbergasted. "Are you serious?! We're in the middle of a fight here!!"
Akiko channeling her father's spirit while protecting her client in "The S Terror".
This needs to be explained. The Sweets Dopants is spitting out sticky goo and about to go for the client. What does Akiko do? She gets a chair and screams, "I WON'T LET YOU HURT MY CLIENT!" and literally hits the Dopants with the chair so hard that it breaks.
Shotaro going hard-boiled in Begins Night, punching out the fake Sokichi/Skull - while untransformed, busting his knuckles open - and delivering a Shut Up, Hannibal! while finally accepting that this Sokichi is a fake, and the real one lives on in his heart.
Akiko in "The F Afterglow". Arms Dopant has put Shotaro and Akiko in a simple sadistic trap. Akiko's bloody hands as she desperately tries to keep Shotaro from falling says it all.
Fang's last second rescue when her grip finally slips, followed by Philip pulling a Big Damn Heroes moment.
For Double's suit actor. Watching as Shotaro, trapped in Phillips body with his crazed partner, tries to use the Joker half to restrain Fang's fury, makes you truely appreciate the skill it takes to convey the two-in-one Rider's nature. Even something as simple as using the right (Fang) side of the body more agressively conveys volumes about the state of minds Double is in.
Accel chasing down the Ice Age Dopant in Bike Form over street. Ice, hills, and even a rollercoaster.
"The D was Seen" saw Philip do the impossible and outwit Weather, denying him the Invisible Memory before bailing out Terui with FangJoker before coordinating a Double Maximum Drive that reversed a gigantic tornado.
The first appearance of CycloneJokerXtreme certainly counts. Especially since his finisher uses four Maximum Drives.
Which sadly did not figure into the CMoA that happened during the real battle against Terror. Trial got RULED by the Terror Dragon so he had to improvise a way to fight it on other terms, so he brings in the HARDBOILDER's Turbuler attachment. Cue him using Double's bike attachments to supplement his original mode to great effect.
The fight at the end of 46. If you weren't pumped up, you are a zombie.
Shotaro overcoming the effects the Terror memory and stopping the Gaia Impact from happening and saving Philip as well.
Shotaro saving Wakana by himself in episode 48. He uses Soukichi's hat to stop Utopia's attack and uses the Memory Gadgets as a distraction, all while W-B-X is playing in the background.
Let's not forget the final climatic battle with CycloneJokerXtreme vs. Utopia, also doubles as a Tear Jerker when you realize it's also Philip's final dance, and at the end of the battle he's going to disappear forever.
Further building on the Battle, Double burns his fingers into Utopia's arm. I repeat, he grabs his arm so hard, that that the fingers are burned into his arm!
Shotaro wearing Sokichi's hat in that episode. After leaving it on the wall for the entire series, he's finally chosen to accept his role as Sokichi's successor.
Every single moment of Kamen Rider Joker in the last episode.
Not to mention Phillip coming Back from the Dead, then working with Shotaro to become Cyclone Joker once again to finish off the Energy Dopant, as the final credits roll to the series theme WBX.
Let's not forget the very last lines of the Double TV series:
Phillip: You haven't forgotten the Catch Phrase, have you Shotaro?
Shotaro: Of course not, Phillip! All those who make this city cry shall forever hear our call:
The villain has essentially won, BUT the entire city gathers at the site of the battle and chant "KAMEN RIDER" over and other again until "the winds of Fuuto" give Double their second wind in the form of wings, they unleash their ultimate Rider Kick on Eternal, destroying him forever and utterly destroying the tower they're fighting on.
The sequence of Ryu busting Mook ass armed only with his gun and his fists, then Jinno stumbles his way into the warehouse.
About that, the entire Accel Booster fight:
After Commander sends Akiko falling to her death, Ryu tries to dash over to save her in normal Accel form but can't do it as well due to his right leg being essentially shattered from a previous fight. Upon obtaining the Gaia Memory Enhancing Adapter, he activates his Booster form in a similar way to Trial. Cue jet engine noises as he flies to Akiko and catches her. Thus begins the fight between Accel Booster and Commander in an absolutely amazing climax. Oh, and he dehenshins in time to walk away suitless as Commander explodes behind him.
Katsumi Daido/Kamen Rider Eternal and NEVER vs the Eyes Dopant. After seeing what the Eyes Dopant did in the V-Cinema, it is satisfying to see how the latter got his ass handed to Katsumi. After the Dopant kills some people Katsumi helped escape, Katsumi releases an insane laughter and delivers a very powerful Death Glare. Then he calmly admits his defeat and tells the Eyes Dopant that he forgot everyone is a devil. This moment alone tells the viewer asskicking is about to happen.
And the fight against Jun Kazu/Utopia Dopant before. After Katsumi transformed, he CURBSTOMPED Kazu to death (literally).