Top attractions in Rome

Are you counting the Vatican in your 5? If so, here are mine in no order

St. Peter's, especially the dome climb
Pantheon
Forum
Capitoline Museum (substitue Borghese gallery if you prefer baroque art to antiquities)
San Clemente underground layers

That was a focus on SIGHTS...but really a top attraction in Rome is...Rome. Wandering the streets, coming across neat squares, coffee, pizza, gelato. I especially like wandering the Trastevere and Monti neighborhoods

Depending on what you mean by "the Vatican" , you are really left with 2 days. The Vatican Museums , Sistine Chapel, St Peters (with dome climb) will keep you occupied until late afternoon, early evening. On an evening walk you can do Piazza Navona and Trevi Fountain

Colosseum, Forum, San Clemente.

St Peter in Chains for Michaelangelo's Moses ( Near Colosseum)

Not exactly 5, but certainly doable in 3 days and gives you a little time to wander and 'discover' Rome

Don't base your itinerary on someone else's idea of a 'must-see'. You could easily spend three days just strolling around the different neighbourhoods and parks, stopping for leisurely lunches, and catch some of the main sites on the way, such as the Spanish Steps, famous piazzas, the Pantheon, Trevi Fountain etc.
To add my two cents worth, Ostia Antica, not to everyone's tastes.

I was just there and we had 3 days, we were able to see quite a bit. But not everything.
You already mentioned Vatican City so I won't list St. Peter's but that was awesome and on my list for sure.

My personal favorites were:
Pantheon
Coliseum/Forum
Trevi Fountain (see it twice-at night and during the day, such a different look!)
Piazza Navona (this can be done at any time)
Borghese Gallery

The cool thing is that in a 2-3 hour walk, you can see many of the top sights (Pantheon, Piazzas, Trevi, Spanish Steps etc), just do Rick's Heart of Rome walk! :)
I also agree that just wandering and eating gelato/pizza etc is one of the main attractions of Rome.

Have fun!

Vatican Museums
Domus Aurea (open only during the week ends, mandatory reservations)
Pantheon
Borghese Gallery
Ara Pacis Museum.

Baths of Diocletian
Villa Torlonia (Mussolini's wartime abode)
St. Peter-in-Chains
Tiber River
Random tabacchi shop/caffe joint

Yes, this is somewhat tongue-in-cheek, but the truth of it is that Rome is so much more than the 'attractions'. At least in the shoulder/off-season, it's a vibe, a pulse, of Romans working, riding the bus, and seemingly enjoying their lives in a historic locale. When we were there, the natives we spoke to were so helpful & nice--unlike other tourist-laden cities in Europe.

Maybe I caught them on a good day (week), but I'd like to think differently...

• A long ramble on the Appia Antica
• Galleria Borghese and Villa Borghese (do together)
• Churches (pick one of any): Santa Sabina; Santa Maria Maggiore; Santa Maria del Popolo; San Giovanni in Laterano, Pantheon, Santa Maria in Trastevere, San Clemente...
• Walk at night to see the monuments, fountains, piazzas, and whatnot lit up
• Get intentionally lost. Eat gelato and drink cappuccino and beer along the way. :O)

6 The church of San Luigi dei Francesi with the 3 Caravaggios still hanging in the Contarelli Chapel, where Caravaggio put them.

7 The Quirinale Palace

8 Saint Peters to see Michelangelo's Pietà

9 Palazzo Massimo alle Terme, to see the bronze Boxeur and the famous ivory mask. One of the very few ivory sculptures left.

10 the Roman houses under Palazzo Valentini (but the Domus Aurea is better: Indiana is still working inside Nero's house!)

I concur with Joe's 06/19/17 post. One caveat: if you would normally choose to climb St. Peter's dome, you might opt out if temperatures are high. I was in Rome last week with 90 degree temperatures. We chose to see the Vatican museum and St. Peter's but skip climbing St. Peter's dome because: 1. the line to climb the dome was long and in the sun and 2. concern that temperatures would be high in the dome itself.

I did do the dome climb (end of June) and inside the dome while climbing had no issue with the temp.

The exterior stone wall is nearly a foot thick so is a great natural insulator, and there are open windows staggered along the stairs. Never felt stuffy.

I would not let the weather stop you.