When I first got into dramas, I remember people would always recommend "the classics" (for lack of better word): Boys Over Flowers, Full House, My Girl, You're Beautiful, My GF is a Gumiho, My Lovely Sam Soon, Secret Garden, etc. 

Nowadays, besides the previously mentioned (b/c I refuse to believe they'll ever go out of style), which dramas do you think could enter into this "classics" list?

I would say You Who Came From the Stars and Descendants of the Sun because of their impact both inside and outside Korea. Any other suggestions? 

I would say Kill me Heal me an Reply 1988
Master's Sun is definitely a classic for me!
Healer
Heirs (love or hate, it was hyper hyped)
Faith is also something of a classic, not for me, but for many people
Reply 1997 and 1988
All the Reply dramas
Bridal Mask
Gu Family Book
Descendants Of the Sun
Missing You
Pinocchio
I Hear Your Voice
Secret
Mask
Rooftop Prince
She Was Pretty
Signal, White Christmas, Gu Family Book, Good Doctor
I would say Healer, White Christmas, Kill Me Heal Me, My Love From Another Star, I Hear Your Voice, It's Okay That's Love and Descendants of the Sun.  I think that these could be considered classics because anytime that I see someone ask for recommendations for themselves or to get someone into dramas, these are some of the titles that they are given (I also recommend these titles).
Healer
The Masters Sun
You From Another Star
DOTS
It's Okay That's Love
Kill Me Heal Me
Reply 1997
Reply 1988
City Hunter
Marriage Not Dating
Fated to Love You (Korean)

For C-dramas - Love Me If You Dare
T-dramas - Just You and Marry Me Or Not
J-dramas - Rich Man Poor Woman
Master's Sun
Princess Hours
Coffee Prince
Delightful Girl Chun Hyang
Signal
Healer
Age of Youth
Cruel City
Six Flying Dragons
Let's Eat
Oh My Ghost
Reply 1988
Fated to Love You