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.
Question
In RPG.SE questions such as this or this, it is clear that stacking proficiency bonuses should be avoided and is "insanely OP."
Why?
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?
Best Answer
It leads to "All or Nothing" skill development
This is going to cause exactly the problem that the 'bounded accuracy' philosophy of 5th edition was designed to address: All skill checks become either trivially easy for experts, or impossible for everyone else.
That is to say, the DM either sets skill DCs low enough for everyone to have a fair shot at making it, in which case the expert almost can't fail (and in that case, why would the non-experts even try?), or the DM sets the DCs high enough for the expert to be challenged, in which case everyone else can't possibly do it (so again, why even roll?).
This leads inevitably to a situation where there isn't much point to basic proficiency; everyone wants to dump all their advancement into a few skills in order to be "the one" for those specific skills, and everyone else avoids those skills like the plague because they know they can't be good at it.
Yes, your idea would allow characters to become more highly specialized -- but that actually isn't a desirable outcome. It isn't fun for one character to be the God Of Investigation while everyone else has to just wait for that one character to handle all the Investigation rolls. Maybe that one guy gets a moment of power-fantasy gloating -- "Ha, nothing escapes my eye!" -- but the rest of the table is effectively being told, "You cannot contribute in this situation." And that's not fun at all.
As a side problem, your concept virtually removes the role of ability scores in being good at a skill. At some point (probably around level 10), the value of the trained skill bonus completely overshadows the ability score's contribution, to the point that it almost doesn't matter if your character is agile or not, charismatic or not, all that matters is what skills you trained up.