Origins: Leliana, Alistair and Morrigan.
DA2: Merrill, Isabela and Fenris
DAI: Cassandra, The Iron Bull and Varric.
I won't choose the PC, because I've got too many to pick just one, and I don't like playing favourites.
Also, I'm not considering Cullen. Heck, we could just say that Cullen, Leliana and Josephine are part of the "dream team" regardless, as advisors. In which case, I'd take out Leliana and put Zevran in, instead.
Almost everyone is just picking their favourite characters from each game, much harder to just pick 3 characters from across all three games imho.
1. Loghain
2. Dorian
3. Sera
Lead by my rogue qunari quizzy Gaston.
That third spot was hard to pick o.o Also now I'm debating switching Sera out for Shale. It is way too hard XD
DAO/A: Leliana, Mhairi, Shale
DA2: Aveline, Bethany, Merrill
DAI: Cassandra, Cole, Josephine(for when we can solve things peacefully which I prefer)
Leader: Inquisitor
All the companions are female except for Cole, who has no gender... Coincidence?!
Almost everyone is just picking their favourite characters from each game, much harder to just pick 3 characters from across all three games imho.
Hero of Ferelden, Alistair, Justice, and Isabella would be my overall picks.
DAO/DAOA: Alistair, Justice, and Shale
DA2: Aveline, Isabella, and Sebastian
DAI: Blackwall, Cassandra, and Sera
Pffffffff so hard. I would have different teams for each of my characters. But I'd have to go with Adrienne Hawke, sarcastic Mage. And her team would be:
DAO: Shale, Morrigan and DAA Anders.
DA2: Varric, Isabela and Aveline.
DAI: Solas, Dorian and Cassandra.
All the companions are female except for Cole, who has no gender... Coincidence?!
Seriously though, it is just a coincidence. I just like their characters more than the others in their respective games.
I guess just everything from Origins to be honest.
For the leader. It would be my Female Dalish elf warden
My main group would be Leliana, Sigrun and Sera. there'd be lots of fun banter going on in that group.
The picks for all three though.
DAO Leliana, Sigrun, Dog
DA2 Merrill, Varric, Bethany
DAI Sera, Cassandra, Dorian
My mage Warden with
DAO - Alistair, Morrigan Leliana
DAA - Anders Nate Velanna
DA2 Aveline, Fenris, Varric, Merrill
DAI - Cassandra, Varric, Dorian, Bull
since you didn't specify which games 
my Jedi from kotor2, MasterChief, Dog from DAO, all led by my circle elf mage warden from DAO (who found the blight cure after dai with Morrigan's help).
since you didn't specify which games
Ah, just for that one you win 
My Sith-ish Exile and My Warden would probably become the most miserable best buddies ever
Rolling with the Inquisitor (Love my Warden but they've got other things on their mind atm). As for companions ... think I'll avoid plot critical ones for the most part as their all doing something important at the moment. They grew up, can't be adventuring all that much these days.
DA:O) Shale, Ogrhen (Love Sten to death but he's an Arishok now, he can't be gallivanting around the world any more).
DA2) Merrill, Varric, Aveline
DAI) Dorian, Cole, Vivienne, Sera
I usually mix-and-match throughout all three games, so no preference on that. However, I do have a dream team that I've always wanted...
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My Inquisitor- Male. Dalish Elf. Sword and Shield warrior.
Leading:
No tactical reasoning. Just want to see what would happen with these band of misfits and how much **** they end up putting themselves in. It's fun picturing my Dalish elf trying to control Zevran from flirting with everyone, Dorian from insulting everyone's terrible outfits (rightly so), and trying to stop Fenris from killing both Dorian and Zevran (not sure which one he would want dead more, hmm...).
Whatever the case, all I know for certain is:
Dorian+Zevran+Fenris= An awesome party for the most charming! There will always be drama, but you'll never be bored. 
I usually mix-and-match throughout all three games, so no preference on that. However, I do have a dream team that I've always wanted...
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My Inquisitor- Male. Dalish Elf. Sword and Shield warrior.
Leading:
- Dorian (romanced by my Inquisitor)
- Fenris (romanced by my Hawke)
- Zevran (romanced by my Warden)
No tactical reasoning. Just want to see what would happen with these band of misfits and how much **** they end up putting themselves in. It's fun picturing my Dalish elf trying to control Zevran from flirting with everyone, Dorian from insulting everyone's terrible outfits (rightly so), and trying to stop Fenris from killing both Dorian and Zevran (not sure which one he would want dead more, hmm...).
Whatever the case, all I know for certain is:
Dorian+Zevran+Fenris= An awesome party for the most charming! There will always be drama, but you'll never be bored.
I'm fairly sure it would be Dorian. A Mage, from Tevinter, on his way of becoming a Magister... Yeah I think he'd go for Dorian first. He might just even give them all the finger and be like: "Nope. I'm not going to be in a party with -him-."
Imagine that party selection screen. *Clicks Fenris, Zevran and Dorian.* *Fenris flips you off as Dorian is added and drops out of the party and becomes greyed out.* *Removes Dorian from party, Fenris becomes available again.*
Hmm, I would say the leader of my group would be my rogue duel-wielder Hawke even though but died in my Inquisition playthrough! So instead, my Adaar tank can lead them. As for the rest...
DA:O - Alistair, Sten, Morrigan
DA2 - Aveline, Merrill, Varric
DAI - Vivienne, Blackwall, Cole
I only picked people who were alive AND stayed in my parties through my main world state (That means goodbye Wynne, Leliana, Zevran, Isabella, Sebastian, Fenris, Carver, Solas, Divine Cassandra and Iron Bull) I just realised how many people in my main world state die or leave or don't get recruited, its quite a lot! Well, thats what happens when you roleplay...
I'm fairly sure it would be Dorian. A Mage, from Tevinter, on his way of becoming a Magister... Yeah I think he'd go for Dorian first. He might just even give them all the finger and be like: "Nope. I'm not going to be in a party with -him-."
Imagine that party selection screen. *Clicks Fenris, Zevran and Dorian.* *Fenris flips you off as Dorian is added and drops out of the party and becomes greyed out.* *Removes Dorian from party, Fenris becomes available again.*
Ha! Ha! That, right there, would be worth the pre-order. Heck, that is worth 10 pre-orders. (Now, I really, really want this to become a reality...this is pure gold of awesomeness!) 
You hear that Bioware? Make this happen, and take all of my monies!
DAO: Leliana, Morrigan, Shale
DA2: Varric and .... I've got nothing (I despised many of the companions and found the rest uninspiring)
DAI: Sera, Cassandra, Dorain.
Led by fun-loving FemQunQuisitor 2H Warrior Rebekah Adaar
DAO - Velanna, Ariane, Leliana
DA2 - Merrill, Isabela, Aveline
DAI - Sera, Cassandra, Briala
Leader - Fem - Inquisitor (Dalish Mage) just because of her former position as the Inquisitor. Without the mark, the HoF is certainly more powerful IMO.
Tabris, Isabala, Tallis and Sera.
Brosca, Alistair, Leliana, Sigrun.
Cousland, Morrigan, Aveline, Nathaniel.
Because I pick teams based on how fun it'd be to picture how long it takes them to realise, and how they'd probably all react, to figuring out just how bad their RWBY-esque acronym would be. At least the top one would probably be amused by it! 
DAO - Velanna, Ariane, Leliana
DA2 - Merrill, Isabela, Aveline
DAI - Sera, Cassandra, Briala
Leader - Fem - Inquisitor (Dalish Mage) just because of her former position as the Inquisitor. Without the mark, the HoF is certainly more powerful IMO.
Don't think Briala is a companion or advisor
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