I think overall, structure is really all about progression. The typical intro/verse/chorus/verse/chorus/bridge/chorus/chorus/(outro) structure does this perfectly, while adding the element of repetition (and thus familiarity) perfectly.
1 - The intro, which potentially introduces the theme of the chorus sans-vocals right from the start
2 - Verse 1 takes the energy down and sets the stage for the chorus
3 - First chorus sets up the eventual payoff at the end
4 - 2nd verse drives the lyrical story and gets you excited to hear the chorus again
5 - 2nd chorus reaffirms your understanding of the song and builds comforting familiarity, yet also just borders on a little too much of the same, which perfectly sets the stage for...
6 - The bridge, which gives melodic variation to the whole song, and builds the final energy towards the...
7 - Final double-chorus, which you should know by now and should be singing (or at least humming) along to (optionally, if the bridge didn't vary the melody enough, this would be a good time for a full-step-up key change, although this was overdone in the 90's
8 - (optional) Outro, possibly to tie in with the intro and give the song a bookend feel.
That said, "the pop formula" isn't the only way to write songs that progress and repeat, it's simply the most obvious. Subtlety in structure is a huge part of what separates different genres/subgenres, and can even define the overall character of a band. For my music, I've always been a fan of repeating melodic or rhythmic themes that are often noticed only subconsciously during the first few listens, due to slight (or not-so-slight) variation in tempo, mode, or arrangement. When done properly, this tends to give each song its own character, which can be somewhat difficult in the context of a 40-60 minute metal album (which very often has little-to-no variation in the instrumentation). One of my favorites is introducing a minor-mode melody line, and repeating it later in the song in a major mode, often with a half-time rhythmic variation in the drums and an additional major-mode melody line.
It's interesting; a lot of my views on songwriting stem from a fictional quote by Robert McKee in the movie "Adaptation" on script/story-writing:
"I'll tell you a secret. The last act makes a film. Wow them in the end, and you got a hit. You can have flaws, problems, but wow them in the end, and you've got a hit. Find an ending, but don't cheat, and don't you dare bring in a deus ex machina. Your characters must change, and the change must come from them. Do that, and you'll be fine."
...which you can easily adapt to songwriting, if you're a little creative:
"I'll tell you a secret. It's the ending that makes a song. Wow them in the end, and you got a hit. You can have flaws, problems, but wow them in the end, and you've got a hit. Find an ending, but don't cheat, and don't you dare slap on a haphazard, out-of-nowhere "epic" ending, with the expectation that it'll make up for the lack of structure preceding it. The character of the song must change, whether in a lyrical sense, or a musical sense (or both), and the change must feel like a logical progression. Do that, and you'll be fine."
Not a perfect translation/adaptation, but I've found it has helped me write songs that I like.