The moment in the originals when you're surfing across from New Bark Town, then set foot on Route 27. What do you hear?
NPC: Congratulations! You've taken your first step into Kanto.
This is the moment that says "you've come this far, keep going", even in the remakes.
In HeartGold and SoulSilver, you can let any of your Pokémon out to follow behind you. Any Pokémon. What more needs to be said?
Having Arceus be your wingman on your journey is both awesome and a little funny, when you realize that only Gameplay and Story Segregation is preventing all those trainers that challenge you from taking one look at the Physical God walking behind you and running for the hills in sheer terror. By all rights every criminal grunt, first-form pokemon, and even most non-Gym trainers should be running from you, but it wouldn't make for good gameplay.
The scene in Gold/Silver/Crystal where Lance storms Team Rocket's headquarters with his Dragonite. (Granted, you help him a bit, but really, Lance could have easily taken care of himself in there.)
Extra Badass and Nightmare Fuel points for apparently telling one of his Dragonites to Hyper Beam the front store's manager. At first you might think the man's pokemon wasn't shown for some game limitation reason, but then when you speak to the guy, he sounds like he just got his ass kicked. Which he DID. By the most destructive Normal-type move in the game: Hyper Beam's base power is 150, beaten only by the suicide moves Selfdestruct and Explosion. You also run into him later having just done the same thing to a Rocket Grunt, who similarly expresses signs of a brutal beating. And just to further confirm Lance has full authority to sic his pokemon on humans, his Dragonite saves you from getting tag-teamed by Executive Ariana by slamming her Grunt sidekick away from you. Lance very clearly channels his more savage manga counterpart while he's in that building. And the Rocket Grunts are terrified of him as a result.
So, you've trained your Pokémon up really far in G/S/C/HG/SS, and you go up Mt. Silver Cave. There's a Trainer at the top:
Red: .............. [battle begins]
Game: PKMN Trainer Red would like to battle!
Also known as one of the biggest Oh Crap moments for many players. It's your former character from Gen 1, sporting all three starters, the Snorlax from Silence Bridge, and either the Eevee you got in Celadon evolved into the game's second best Psychic in Gen 2, or the Lapras you were given for saving Silph Co in Gen 4. And to top it off, it's pseudo-Ash!Red from Yellow Version, with a Light Ball-toting Pikachu that is the highest leveled trainer pokemon in the series (81 in GSC and a terrifying 88 in HGSS) at the top of his lineup, who can probably nuke half your team on his own.
If you talk to your walking Pokémon when in the Pokémon League in HG/SS, they'll sometimes give you a slow nod as if to say "we came this far, let's finish this". Every Pokémon does this, but it's a CMOA when your starter, which you've likely used and had follow you through the majority of the game, does this. Even more if it does this a few steps from the Champion.
Alternatively, sometimes your Pokemon will let out a battle cry instead. Awesome either way.
In HG/SS, after you've beaten him in Mt. Moon, Silver goes to the Dragon's Den to train. If you visit him there, you and Silver have a tag team battle against Lance and Clair.