Emilia Clarke aka Daenerys Targaryen full name and full title:

Daenerys Stormborn of House Targaryen

Trueborn Queen of the Andals and the First Men

Khaleesi of the Great Grass Sea

Breaker of Chains

The Unburnt

Lady Regent of the Seven Kingdom

Protector of the Realm

Queen of Meereen

Mhysa

The Queen Across the Sea

The Silver Queen

Khaleesi of the Dothraki

Lady of the Seven Kingdoms

Rhoynar

First of Her Name

Mother of Dragons

Did I miss any? Did I get the order right?

Some explanation:

Tim asks, “Can you maybe enlighten us on the backstories for some of the titles we keep hearing in the show? (E.g. I get “Mother of Dragons,” but should I know “Storm-born?” Or is that just the Westerosi equivalent for Targaryens of how dudes named Murphy will always be called “Murph?”)Daenerys’s titles:

Stormborn: Literally, she was born during a storm. A bad one that wrecked the remaining Targaryen fleet after her mother’s flight from King’s Landing in the wake of Robert’s Rebellion. I wish we did this in our world, like, “Tim, Overcast Day with a Light Drizzle-Born.”

First of His/Her Name: First King/Queen with whatever specific name they already have.

The Unburnt: Daenerys survived the flames of Khal Drogo’s funeral pyre at the end of Season 1 without a scratch. Totally worth bragging about.

Khaleesi of the Great Grass Sea: The title of Khaleesi refers to the wife of a Khal; the Khal is the leader of a Dothraki tribe. I think it’s fair to assume that there are other Dothraki tribes roaming the Dothraki Sea who would disagree with the “Great Grass Sea” modifier used here.

Breaker of Chains: She steals rappers’ chains.

King/Queen of the Andals and of the First Men: Westeros has been the site of many migrations throughout its history. The First Men refers to the first human inhabitants of the continent of Westeros, who, roughly 12,000 years ago, crossed over from Essos into Dorne via a land bridge that existed between the two continents, and gradually colonized the whole land mass. Anywhere from 2,000 to 6,000 years after the arrival of the First Men, the Andals began migrating over from Essos by sea. The Andals brought the Faith of the Seven with them, and eventually their culture became the dominant culture of Westeros. The Andals displaced the First Men from most of the continent except the North and lands beyond the Wall. Generally speaking: People of the North and beyond are most likely ethnically and culturally North men, while everyone else is ethnically Andal. (In the books there’s a third cultural and ethnic group called the Rhoynar, but it seems as if they don’t exist in the Throneiverse.)

Season 4 episode 10: Added Queen of the Rhoynar. See below.

Some of these, like The Unburnt and Queen of Mereen, would make great band names.

You're right!

Going through the list, almost any of them would make fine band names:

http://pandawhale.com/post/37974/candida-and-the-veggies-would-be-a-great-rock-band-name

"Ladies and gentlemen, we are Breaker of Chains! Good night!!"

There's also Jorah Mormont, Lord of Friendzone, first of his name and keeper of tampons.

Jorah Mormont, relation to former Lord Commander of the Wall Mormont and to current little girl Mormont leader of Bear Island who, we learned in the last episode, sent a note of Stark loyalty to Stannis the Mannis.

Found that here:

http://reddit.com/r/gameofthrones/comments/33rw87/s5_predictions_discussion_503_high_sparrow/

Is this pic from S4:E10?  I wondered when pleat technology was born.  :)

I wonder, too. The pic is from May 24 so it's either episode 7 or 8 of season 4.

Oh, in episode 10 Daenerys added a new title:

Daenerys had a new title added, “Queen of the Rhoynar.” Who are they?

The Rhoynar are an ethnic group originating from the banks of the Rhoyne River in Essos. Roughly 700 years ago, the expansion of the dragon-equipped Valyrian Freehold forced the people of the Rhoyne to flee their ancestral homes. Naturally strong seafarers, the Rhoynar set out for Westeros led by their warrior-queen Nymeria (the inspiration for Arya’s wolf’s name), where they eventually became part of the region’s cultural flavor. Much of Dorne’s signature licentiousness and relative gender equality come from the Rhoynar. The Rhoynar are one of the three main ethnic groups in Westeros, along with the First Men and the Andals.

I like this outfit from season 5, too:

Found that fabulous dress in season 5 episode 7:

Reddit comments:

https://reddit.com/r/gameofthrones/comments/389mzh/season_5_daenerys_dress_in_episode_7_was/

Top comment: "I imagine that staying clean for a whole 30 seconds."

Who wore it better?

They could be sisters! Sorry Gwyneth; Daenerys wore it better. 

Daenerys vs Tyrion titles:

http://pandawhale.com/post/41952/daenerys-vs-tyrion-titles-they-call-themselves