In general, things don't stack because the PHB specifically indicates the starting point:
Dragon Ancestor (PHB p.102, emphasis mine):
At 1st level, you choose one type of dragon as your ancestor. The damage type associated with each dragon is used by features you gain later.
You can speak, read, and write Draconic. Additionally, whenever you make a charisma check when interacting with dragons, your proficiency bonus is doubled if it applies to the check.
Expertise (PHB p.54, emphasis mine):
At 3rd level, choose two of your skill proficiencies. Your proficiency bonus is doubled for any ability check you make that uses either of the chosen proficiencies.
At 10th level, you can choose another two skill proficiencies to gain this benefit.
Neither of these modify your proficiency bonus, only what is done to the roll.
- dragonAncestorBonus = proficencyBonus × 2
- expertiseBonus = proficencyBonus × 2
Since both start with your proficiency bonus, you get to choose:
- total = roll + proficencyBonus
- total = roll + dragonAncestorBonus
- total = roll + expertiseBonus
To help make things more clear, the PHB states (p173-174):
Your proficiency bonus can't be added to a single die roll or other number more than once. For example, if two differenct rules say you can add your proficiency bonus to a wisdom saving throw, you nevertheless add the bonus only once when you make the save.
Occasionally, your proficiency bonus might be multiplied or divided (double or halved, for example) before you apply it. For example, the rogue's Expertise feature doubles the proficiency bonus for certain ability checks. If a circumstance suggests that your proficiency bonus applies more than once to the same roll, you will add it only once and multiply or divide it only once.
To get around this constraint, the ability would need to be similar to "increases Persuasion proficiency by 5" or "adds 4 to all persuasion checks."
For the most part, you will not find major stacking effects in D&D-5e. If you think you found something which is broken or too good to be true, it probably is.