Some Factors to Consider
The actual math is well covered by other posts, but I think there are a few further factors to keep in mind. It may be that balance-wise these factors make the proposed rule more in line with your goals, so this is not meant to be critical so much as simply point out instances where it makes a substantial difference.
What is Your Primary Goal
Your statement that "this way players can stack +1 for every clever thing they do" seems to indicate that you perhaps are focused on situations where the DM gives discretionary advantage to people for a clever plan, good roleplaying, etc. How often this applies depends entirely on the DM, and with a sufficiently generous DM may make this a very strong approach to advantage in many situations. It would essentially just make advantage a matter of the DM naming a modifier they will be give based on how many ways they think the player has been clever in their plan or what not. This seems like additional work on the part of the DM rather than just being able to say "you have advantage" if there is any advantage-worthy aspect, but it would give the DM more leeway to determine what is fair. Of course, they can already alter the DC to accommodate factors the players may initiate through their approach to the task, so this really makes little difference if the DM is comfortable making such adjustments.
If you are a DM for whom awarding people for clever approaches to skill checks is the primary goal with this house rule, then I would strongly recommend simply adjusting DCs based on their approach, and perhaps not even granting advantage in these situations if you don't like double rolls. It would then have no balance issues beyond this.
Multiple Advantage Would Be More or Less Plentiful and Important For Different Sorts of Circumstances
There are many situations where consistent advantage is dictated by the rules, namely various class and feat features, various spells, items, and the help action.
Since your proposed form of advantage is weaker under most circumstances and the class features (barbarian reckless attack for example) and feat features (shield master for example) are mostly major mechanical aspects of the given class, subclass, feat, etc. this can radically weaken these character build options. These are mostly combat oriented sources of advantage, and opportunities to stack advantage in combat will be generally less frequent because there are simply too many other uses for peoples actions, bonus actions, etc. in combat. Perhaps the barbarian suffers most on this front since reckless attack is a major class feature and brutal criticals are also a major class feature which is twice as easy to trigger when granted advantage (criticals are explained below).
Spells, such as fairy fire, are one of the other major sources of automatic advantage (and disadvantage), particularly in combat. This radically changes the value of these spells. Particularly for low level casters for whom advantage granting spells are one of their strongest potential contributions to combat this is a substantial change, and further so if applied to disadvantage as well. The bard in particular comes to mind, as her primary damage dealing cantrip is vicious mockery, which is weak on damage but causes disadvantage, and for whom direct damage dealing spells are fairly rare until she gets to raid other class's spell lists at higher levels (not until level ten for non-Lore Bards).
Some items, such as the Boots of Elvenkind also grant advantage on something. You would have to consider whether you, effectively want to make them just +1 items on some sort of roll or not.
Finally we have the help action. Being able to grant this as a bonus action is one of the key features of the Mastermind Rogue (I would argue that it is the only really strong feature they get in the first two tiers), so we have another class who is now balanced much differently. Being able to grant it through a familiar is a major aspect of being a wizard as well.
Beyond this taking the help action in combat is something most players do but rarely, and certainly many party members spending combat actions on help seems unlikely so heavily stacked advantage from this is going to be fairly rare. Meanwhile in the case of many ability checks there is no reason the whole party would not each help the one player make the roll if the DM allows. The significance of this will depend heavily on how amenable the DM is to letting multiple people try the same skill check and to letting players use help in these circumstances.
There is also the aspect of the help action that it often involves one player giving up there roll so another player gets two rolls. Whatever the mechanical strengths this both makes it a more straight-forward choice for the player and for some players a source of camaraderie.
Natural Ones and Twenties
If you are only rolling one die than in combat Crits are half as likely, which is a further reason that this is substantially less powerful. This is a particularly important thing to keep in mind in terms of the balance of classes or subclasses for whom a substantial benefit revolves around the crit, such as a barbarian or a champion fighter. A critical failure would be more likely, since you have half the chance to avoid one, but these are far less important since usually one is just part of a range of numbers that will obviously not hit, however this is also tied to the a few mechanics such as the halfling "luck" and would effect the balance.
Also consider that at many tables the principle that a natural twenty or one leads to a dramatic critical failure or success on ability checks regardless of modifiers (this is quite possibly the most common house rule). For advantage this halves the chance of such a critical success and doubles the chance of critical failure.
Applying the comparable rule to disadvantage, if you adopted that as well, would of course have the inverse effect for all of this.
Advantage AND Disadvantage?
One aspect of advantage and disadvantage not stacking is that the cancel each other out completely no matter how many more sources you have on one side or the other. You would also have to take into account how you would handle this rule.
I am presuming that you intended this rule to apply to all advantage rather than just your houserule additional advantage. If that is not the case than the significance of this rule is quite different and much, much lower for the very reasons that have been explained in this and several other answers. It would simply be a (usually) modest buff (or if you apply it to disadvantage debuff) to rolls, and, unless you let whole party take help actionS on an out of combat skill check, doesn't seem like it could really break any aspect of the game.