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

Jeff Beck - Blow by BlowBill Frisell - Gone Just Like a TrainLittle Feat - Dixie Chicken, Sailing Shoes
Fleetwood Mac - Then Play OnEagles - On the Border
Wes Montgomery - Smokin' at the Half NoteFrank Zappa - Hot RatsMichael Brecker - Two Blocks From the EdgePat Martino - Live at Yoshi'sJohn Scofield - A Go-Go

I think what the OP meant by "Album" was musical work, not necessarily the vinyl itself. I'm sure must of us play our records all the way through.

Quick story: My neighbors come over every now & then for foosball, beers, and jams. I used to just load up the cd player and put it on random. This was about 10-12 years ago before vinyl made its resurgence. My buddy came out of my "storage closet" of music with AC/DC's Back in Black LP and asked if I could play it. I almost laughed because I thought - "Everyone knows that!". 

But that was the point, everyone knew it. Everyone knew exactly what was coming up next after each song and totally dug it!

That experience opened my eyes and I've never used random since, even on cd's. 

To the OP, I clicked on this as one of the “posts of the week” to add my picks only to find my first two picks on your list already! - “DSOTM” and “I Robot”.

In fact, I was actually listening to “I Robot”, having just “rediscovered” it today (no lie!) and had just about let the thing run all the way through when I came across the Audiogon Weekly email. For those who enjoy this album, I urge you to look for the “Legacy Edition”. I think it’s been remastered and loads of studio track extras and commentary. I love streaming! Would have never found this version ow.

In addition to the two above I’d add Al Stewart’s “Year of the Cat”. (See a theme emerging here?), Genesis “Trick of the Tail” and several others. All from the era of Rock Opera and themed albums. Not many cohesive themed albums that tell a story these days. All sound bites or hits.
  
Oh, and Genesis “Lamb Lies Down on Broadway” MUST be listened to in its entirety.

I generally listen to the entire album unless it has (in classical music) a particular piece worth listening to and not the other(s).  Often one side will sound superior to the other side of the record (factual statement as to pressing differences-see Better Records for explanation).  For vocal albums, I often pick and choose which selections to hear.  I have 25,000 LPs.  For 78s, I listen through to the 3 to 5 minutes per side obviously (7,000).  I tend to listen to entire CDs as well (7,000) even though they can last in the low 80 minutes (classical collections from Decca).   

The Who. Who’s Next
Elaine Stritch At Liberty 
Beatles Rubber Soul
Stones Sticky Fingers
compilation cds don’t count. That’s cheating......

Does anyone remember the “Perfect Album Side” on WNEW FM?