I find a society where everyone strives to do something that society would consider "profoundly incredible" to be remarkably stable. Certainly far far more stable than a society where everyone strives to be as rich as possible, or any similarly mundane goal.
What makes this goal so powerful is the power of strange loops. Strange Loops are a construct originally put forth by Douglas Hofstader in the amusingly titled book, I am a Strange Loop. The idea of a strange loop is that one goes up a hierarchical chain only to get back to where we started. In this case, the loop occurs because the phrasing assumes society has goals (which permit something to be "profoundly incredible"). However, society's goals are defined by the accumulation of our own goals. If our goals are based off of society's goals, we've come full circle!
One way to break this strange loop is to say that one strives to do something that they perceive society would consider "profoundly incredible." This breaks the strange loop because it is possible for one's perceptions to be quite out of whack with reality. An evil genius may actually believe that the world wants sharks with [freaking] laser beams on them. A world such as that described in the question has to deal with this.
To deal with these extremes, it's helpful to split up the different kinds of worlds one might live in into categories, and observe how one's actions might change those worlds. The first category I'd explore is a de-coherent world. This is a world where everyone's perceptions of what society wants are in different directions, generating no cohesive point of view for society as a whole. In this extreme, you find those who strive for something society would consider "profoundly incredible" will begin making their ideas a reality. Picture a world full of Nikolai Teslas. Of course, not everyone will succeed at being profoundly incredible on this global scale, but there's a really neat cheat that appears. What if one shrinks one's world? There's no reason one would choose to do so, because that clearly cannot lead to profound incredibleness, but it may happen by accident. If it does happen, a curious thing happens: this smaller world may be more coherent than before because it has excluded some viewpoints. This can lead to one being happy in their own little world. And, in fact, this smaller world may encourage one to not widen one's perspectives to return to the previous world size. We'll come back to this later.
The next extreme is a highly coherent world. In this world, there is a very clear viewpoint for society from which "profoundly incredible" can be defined. This viewpoint leads to a mixture of two kinds of profoundly incredible: competitive and cooperative. A competitive world leads to a decrease in coherence as people strive to beat each other, dragging them down towards a decoherent world. Cooperative ideals, however, permit any number of people to seek them at once. Such cooperative ideals are "stable" in this coherent world.
The final extreme is a divided world, with a small number of mutually-competitive worldviews. In this case, each group assumes "they know best." Competition eventually leads to one side coming out on top, leading to a coherent world.
If we consider the latter two worlds, it is easy to see that the worlds stabilize on the coherent category in most cases. Coherence encourages coopoeration, keeping that stable. Meanwhile, most rebellious ideas are crushed unless they learn to adapt them to be more cooperative.
The first world's analysis is more complicated. The world naturally fractures into a bunch of smaller worlds as each person's world view slowly decreases in size. Soon this leads to a large number of coherent worlds that exist in a web, and this web slowly leads all of them towards coherence!
In the end, this shows that a society where everyone seeks to do something society would consider "profoundly incredible" will naturally lead to everyone coherently seeking the same goal. However, that assumption does require there to be cooperative goals in the first place. If the only valid "profoundly incredible" goals are those which require competition, the whole pattern falls apart.
We can find proof that there must be cooperative goals to be had, because each person has their own body and mind. A cooperative goal may call for one to achieve something uniquely beautiful with their own body and/or mind. This cannot possibly be competitive, for each person has their own body to make beautiful. "Be the best you can be" is a valid goal, in such a coherent world. Intriguingly enough, it's also a valid goal in any of the other worlds.
Pretty profoundly incredible, isn't it.