Albums that you listen to straight through / start to finish whenever you put them on!

Here's probably my number 1:  Mary, the solo album by Mary Travers, 1970/71.  

For me it has it all.  Wonderful wonderful singer/voice, great selection of songs, fantastic sounds, engineered by Phil Ramone.  

This has to be one of my top 5 LPs/CDs/Recordings of all time.  

Of course "there is no accounting for taste".  

I just love this record, only wish there were more like it, raises so many emotions in me when I play it, and of course it's a total joy for me to listen to.  

The OP makes a couple of really good points:

1) Albums that you truly want to listen to all the way thru are rare.

2)But,  it's a good idea to listen to the entire album because our tastes change over time. I know that I've rediscovered countless songs that I either forgot about or just didn't appreciate initially. However, I know I'll never change my mind about "Alabama Song" from the debut album by The Doors.

JJ Cale - "Naturally"
Bruce Cockburn - "Inner City Front"
Lloyd Cole & Commotions: "Rattlesnakes"
Crack The Sky - "Crack The Sky"
Michael Franks - "The Art of Tea"
Love - "Forever Changes"
Pousette - Dart Band - "Pousette - Dart Band"
Wishbone Ash - "Argus"

Blue Oyster Cult - "Tyranny and Mutation"
Maria McKee - "Peddling Dreams"
Iris Dement - "Sing the Delta"
Crowded House - "Temple of Low Men"
The Band - "Music from Big Pink"

And one very important pair not to be missed and always played through (NOTE: Not necessarily vinyl, but albums none the less).

Tom Waits - "Mule Variations" 
Tom Waits - "Alice"

Fascinating... I guess I am a rare bird. But, of course, the bird is the word.

I listen to both sides of a record only the very first time and rarely afterwords. I get a sense of the vibe of the side and play one or the other depending on mood (and guests) forever afterword. When a side ends I assess the mood and, generally speaking, I am so inspired by that music I want to explore another record to push that envelope or completely change tack. Sometimes the mood does lead to flipping sides. Sometimes. I did that yesterday once.

Digital is a whole other story but does still includes listening to the full album the very first time. In fact, I now wish Roon had an option to display and play as album sides.