I am the DM for a campaign and play with the following house rules:
When gaining proficiency in any skill (during character creation, taking the Skilled feat, etc.), you may choose a skill more than once. Your proficiency bonus for that skill is the number of times that skill has been chosen times your base proficiency bonus.
When you gain proficiency in multiple skills at the same time, the skills chosen must all be distinct.
Any feat that allows a player to "double" their proficiency bonus for a skill instead allows the player to mark one additional level of proficiency in the given skill (e.g. Expertise).
Example of stacking proficiencies
Create a very sneaky and acrobatic Kenku Rogue.
Through Kenku Training, choose to gain proficiency in Acrobatics and Stealth. Rule (2) prevents me from choosing Acrobatics and Acrobatics. Through the Rogue features, choose to gain proficiency in Acrobatics and Stealth (and any two others) and Expertise in Acrobatics and Stealth. From the Criminal background, gain proficiency in Stealth.
At level 1, the character's base proficiency bonus is +2. The player has selected Acrobatics three times, and stealth four times. The new proficiency bonus to Acrobatics is +6 and to stealth is +8.
In RPG.SE questions such as this or this, it is clear that stacking proficiency bonuses should be avoided and is "insanely OP."
I see a trade-off here where a character could be (unreasonably) skilled at one or two things, but isn't very good at anything else.
Why is allowing players to stack their skill proficiency bonus considered to be overpowered?