In no particular order…
“Public Enemy No. 1” by Public Enemy
“Paid in Full” by Eric B & Rakim
“Disintegration” by the Cure
“My Philosophy” by Boogie Down Productions
“Running to Stand Still” by U2
Yes, rap existed in the 80’s.
Except it was done by After the Fire, of which Falco was a member. 
My picks:
Wait. I can’t restrict myself to just five. So I’m not even gonna try.
Robin
"Take on Me" by A-ha … My very first single (ah, those days of vinyl)
"Carrie" by Europe … My very first party (slow dancing and everything!)
"Reality" by Richard Sanderson … My very first trip abroad (Vive la France - Vive those eyes of Pierre Cosso)
"Suck My Kiss" by the Chili Peppers … My very first make out session (Yes, I know it was 1991, but close enough)
and…drumroll
"I’ve been looking for freedom" by none other than David Hasselhoff himself.
Yes, I am from Germany. Yes, I know the song sucks. But it instantly transports me to the 80s because it was never ever played again under any circumstances. Plus, it reminds me of Knight Rider (see thread Ever discover something you loved really sucks?)
I was beyond teens (not by much) but can I play too?
Promises in the Dark (Pat Benatar)
In The Dark (Billy Squier)
Pour Some Sugar on Me (Def Lepard)
You Shook me All Night Long (AC/DC)
What A Fool Believes (Doobie Bros.)
- Flock of Seagulls - I Ran!
- Kim Wilde - Kids in America!!
- Gary Numan - Cars!!!
- Ray Parker Jr. - Ghostbusters!!!
- Amy Holland - She’s on Fire!!!
- Paul Engemann - Push it to the Limit!!!
Pop songs:
Men at Work - “Down Under”
The Police - “King of Pain”
Peter Gabriel - “Shock the Monkey”
Cyndi Lauper - “She Bop”
Euthrymics - “Who’s that Girl”
And some songs that I still listen to on the way to work:
Skinny Puppy - “Assimilate”
Ministry - “Stigmata”
Front 242 - “Headhunter”
Sisters of Mercy - “Temple of Love”
Public Image Limited - “Rise”
I have way more than 5, can I list like my top 11 ?
Girls Just Wanna HaveFun- Cyndi Lauper
Tainted love - Soft cell
Jessies girl- ?
I run-Flock of Seagulls
Pour some sugar on me- Def Lepard
Take on me - Aha
We got the beat - The gogo’s
Anythng by Bon Jovi
The Amadeus song
Whip it- Devo
Love Shack - B52’s
I could go on forever.
Some of my Brit favorites (favourites) from the eighties:
“In Your Eyes” - Peter Gabriel
“Modern Love” - David Bowie
“Why Worry” - Dire Straits
“The Ghost In You” - Psychedelic Furs
“How Soon Is Now” The Smiths
“Fascination Street” - The Cure
“Wishing (If I Had a Photograph)” Flock of Seagulls
“Sweetwater Pools” - Screaming Blue Messiahs
Some of my American favorites:
“Voices Carry” - Til Tuesday
“Let’s Go Crazy” - Prince
“Whip It” - Devo
“Word Up” - Cameo
“Heartbeat City” - The Cars
“You Dropped a Bomb On Me” - The Gap Band
I could go on, but I’ve went past the original “pick 5” a little bit…
Sonic Youth - Schizophrenia
Pixies - Where is my Mind?
Richard & Linda Thompson - Wall of Death
Nick Cave - The Mercy Seat
Joy Division - Isolation
- Standing In The Dark - Platinum Blonde
- Billy Jean - Michael Jackson
- I Would Die 4 U - Prince and the Revolution
- Don’t You (Forget About Me) - Simple Minds
- Telefone (Long Distance Love Affair) - Sheena Easton
I could have included Duran Duran, Madonna, Sheila E, Huey Lewis and the News, Eric Clapton, Rush, The Pursuit Of Happiness, etc… but this is only a top 5 list unfortunately.
Corey Hart - Sunglasses at Night
Terence Trent D’Arby - Sign Your Name
Kiss - I Was Made for Loving You (a 1979 release, I believe, but I bought the 45 in the 80’s, so there :p)
Underworld - Underneath the Radar (from before they were an electronica band)
U2 - Unforgettable Fire
Pat Benatar - We Belong
(that’s 6, but I can’t drop any one - they’re all equally favourites that I still play to this day)
Land Down Under - Men at Work
We Run - Strange Advance
Roxanne - The Police
And the Days Go By - Talking Heads
Push it - Salt’n Pepper
not top 5… but worth mentioning
The Sun Always Shines on TV - A-ha
Rain - Dragon
Sierra Leone - Coconut Rough
Nightshift - Commodores
Red Red Wine - UB40
This is way way way too hard!!
Belly Of The Whale - Burning Sensations
It Must Be Love - Madness
(Birds Fly)Whisper To A Scream - Icicle Works
Blister In The Sun - Violent Femmes
But Not Tonight - Depeche Mode
To the previous poster who’s UN I didn’t note, Jessie’s Girl was done by Rick Springfield.
This was hard, but that’s my list, in no particular order, and I’m sticking to it. 
Very difficult choice.
Mary’s Prayer - Danny Wilson
Every Little Thing She Does Is Magic - Police
P-Machinery - Propaganda
Chance - Big Country
All Stood Still - Ultravox
I couldn’t vouch for the quality of all of them now, but they probably have the most memories attached.
Der Plan - “Gummi Twist”
DAF - “Ich Will”
Nina Hagen - “TV Glotzer”
Die Doraus und die Marinas - “Fred vom Jupiter”
KFc - “Wer hat Lilli umgebracht?”
And just where were you lot when the 80’s was going on? Hiding behind the door? <thrrrp>
Welcome to the Pleasuredome - FGTH
Rent - Pet Shop Boys
Blue Monday - New Order
A New England - Kirsty MacColl
Somewhere In My Heart - Aztec Camera
Right, that’s you lot told…