How do you raise a dragon's Star Rank in Game of Thrones: Dragonfire?

Star Rank goes up with dragon relics, at the Hatchery. It's permanent and never resets between reigns.

Relic costs per tier: 1★ = 15 (hatching), 2★ = 40, 3★ = 100, 4★ = 180… up to 7★ = 800. The curve climbs steeply.

Stars do 3 things: your dragon grows bigger, gains stats, and above all unlocks new habit slots (roughly one habit every 2 ranks). Star Rank opens the slot; chests then provide what's needed to take that habit from level 1 to 5. It's the most rewarding progress you can buy, because it's definitive.