Older person reporting, born in 1980...
5: Talking Heads - Speaking in Tongues
You may be skeptical, but I was lucky enough to have parents that were into "weird" music in the early 80s and taught me how to read from stereo catalogs and album covers. From as early as I can remember I was listening to Talking Heads, REM, Devo, etc. long before they became "cool".
10: Motley Crue - Dr. Feelgood
Unfortunately, parental influence on my tastes began to wane and I was watching MTV every chance I got. Was big into Crue/GNR/Poison/etc.
15: Smashing Pumpkins - Mellon Collie and the Infinite Sadness
This one pretty much defined the high school years for teenagers of my era. It just perfectly captured all the feelings of growing up and was one of the first albums where I really felt a connection with the music and like there were other people out there feeling the same way as me. It probably sounds a bit cringey to listen to now if you weren't a teenager when it came out, though.
20: Autechre - Incunabula
In college I really got into electronic music. I don't remember one specific favorite album so this could really be any of the older "IDM" albums by Orbital, Autechre, or Aphex Twin. But Incunabula was definitely one that was so different than anything else I had heard before and really expanded my ideas of what music could be. Unfortunately, I tried listening to it recently and it has not aged well at all.
25: Modest Mouse - The Moon and Antarctica
After a 9-year relationship ended, I started listening to different types of music and that's when I really got into indie rock. Modest Mouse, Radiohead, the first albums by Arcade Fire and Interpol, etc. got me through some really dark times. I'd still consider all of them near the top of my favorite albums ever, but a lot of the songs are hard to listen to now because there was so much emotion associated with them back then.
30: My Bloody Valentine - Loveless
It was released when I was 11, but I didn't really hear of it or listen to it until my late 20s. It's just an amazing masterpiece of an album, every song is excellent from start to finish, and it sounded like something created in a different universe. It's ageless, still sounds just as good today.
35: ???
The year is still in progress, so I don't have to choose yet, right? Honestly I don't think my absolute top albums have really changed much over the last 5-10 years. I have listened to a lot of new music that I enjoy but I wouldn't consider anything from the past 5 years on the same level as my all-time favorites. If I had to name one single album I guess I'd have to randomly pick one from the set of {Moon and Antarctica, Lonesome Crowded West, Mogwai Young Team, Frances the Mute, OK Computer, Loveless}.