I would second “A Day in the Life”
Since you were 11.14 months old? 
What made the Beatles special? IMHO harmonies, great guitar riffs, and especially song subject matter way beyond what anyone else was doing at the time. Walrus is wonderful, but it is way too John to be definitive. My three choices, all previously mentioned are
I Want to Hold Your Hand - harmonies, great beat, and risque lyrics “my love I can’t hide???” Perhaps this is the first Beatles song that connected with me.
Day Tripper - Great riff, great vocals, and subject matter no one else could get away with.
Day in the Life - John and Paul working together, but showing their differences.
While The End is also great, I think it has too much self knowledge. They were trying to get back to their roots, it didn’t just happen. They succeeded though.
Yeah, my mum likes The Beatles, so I’ve been singing along since I was knee high to a grasshopper. I’m 19 too. I think a great deal of kids would know who they are.
Wow, I never thought of it that way, and I always considered myself relatively perverted.
Actually, when I first heard that line, I made the same mistake Bob Dylan did, which was thinking that they’re singing “I get high.”
“A Hard Day’s Night.”
Later on, “A Day in the Life,” and finally “Get Back.”
Well, the movie A Hard Day’s Night would be hard to top for even professional actors, which the Beatles weren’t.
At least they weren’t Elvis Presley or Justin and kelly when it comes to film.
Another vote for “A Day In the Life” - it seems like a magnum opus, integrating their weirdness, but not getting too out of hand and running the full gamut of musicianship and studio magic.
I fourth or fifth “A Day in the Life.” It was the first thought I had when I read the OP.
Yeah, “Day in the Life” was actually my second choice. I just wasn’t sure it really showcased George enough. It probably is the quintessential Lennon/McCartney composition, though.
Mrs Geek says that “Norwegian Wood (This Bird Has Flown)” from Rubber Soul gets her vote. George plays the sitar. The Beatles as World Musicians?
The section of the song from “Sitting in an English garden…” onwards is the part which also contains the Shakespeare broadcast. This was added to the recording at random from a live radio feed while they were making the mono mixdown. This would have made it impossible to make a true stereo replication of that section for the stereo mix of the song, as they’d have to be able to get the radio broadcast again and mix it in at the identical locations. However, it was locked in the mono mix. The only alternative available for a stereo mix was to do what you suggested, one channel brighter and the other duller from that point on. There is some subtle panning back and forth of the fake stereo image, but it is, nonetheless, mono.
I could make a case for about fifty, but I want to second the nomination for There’s a Place. Have you listened to it lately? It’s an under-the-radar song. Here they were in 1962 [yeah that’s right '62, cause there’s a live version of it on the German Star Club album] singing:
There’s a place / I can go
when I feel low / when I feel blue
Pretty standard, but then they veer into:
and it’s my mind / and there’s no time [beat]
when I’m alone
And the harmonies on “and it’s my mi-i-i-i-ind” are unexpected and fresh, and the lyric is both earnest, young, and wise, and poignant.
Definitive? Maybe. It was probably one of those songs written “eyeball to eyeball” as Lennon put it. It’s a two and a half minute burst of fresh air, and a signal that these guys knew how to get to that place, and take you there too.