Here are four applications, but some of them are cheating. I'd like to give more examples involving homotopy groups of more general spaces, but will try not to.

(Tagged cw - feel free to add.)

• The fact that $\pi_k(S^n) = 0$ for $k < n$ is used implicitly all the time; this high connectivity means there are few obstructions to embedding small complexes $X$ into $S^n$, and hence having tools like Alexander duality available to study $X$.
• In another question the classification of simply-connected 4-manifolds up to homotopy type is mentioned. This uses $\pi_3$ of a wedge of 2-spheres, so technically it's not what the original question asked.
• The existence of any kind of unital multiplication $S^n \times S^n \to S^n$, such as one that would be obtained by a division algebra structure on $\mathbb{R}^{n+1}$, is equivalent to a certain element existing in stable homotopy groups of spheres (a class of Hopf invariant one). Adams proved that this element is not realized in a homotopy group unless n is 0, 1, 3, or 7.
• The more serious computations of algebraic K-theory groups using topological cyclic homology, such as the K-groups of the integers, often require serious input knowledge from stable homotopy theory. For example, when Ausoni-Rognes computed the V(1)-homology groups of $K(\ell)$, they used knowledge about the stable homotopy groups of spheres to identify certain classes as necessarily survivors of several of the many spectral sequences involved. This is not the deepest part of their work by any means, but is supposed to indicate that this kind of knowledge is "jacks or better to open".

Here are four applications, but some of them are cheating. I'd like to give more examples involving homotopy groups of more general spaces, but will try not to.

(Tagged cw - feel free to add.)

• The fact that $\pi_k(S^n) = 0$ for $k < n$ is used implicitly all the time; this high connectivity means there are few obstructions to embedding small complexes $X$ into $S^n$, and hence having tools like Alexander duality available to study $X$.
• In another question the classification of simply-connected 4-manifolds up to homotopy type is mentioned. This uses $\pi_3$ of a wedge of 2-spheres, so technically it's not what the original question asked.
• The existence of any kind of unital multiplication $S^n \times S^n \to S^n$, such as one that would be obtained by a division algebra structure on $\mathbb{R}^{n+1}$, is equivalent to a certain element existing in stable homotopy groups of spheres (a class of Hopf invariant one). Adams proved that this element is not realized in a homotopy group unless n is 0, 1, 3, or 7.
• The more serious computations of algebraic K-theory groups using topological cyclic homology, such as the K-groups of the integers, often require serious input knowledge from stable homotopy theory. For example, when Ausoni-Rognes computed the V(1)-homology groups of $K(\ell)$, they used knowledge about the stable homotopy groups of spheres to identify certain classes as necessarily survivors of several of the many spectral sequences involved. This is not the deepest part of their work by any means, but is supposed to indicate that this kind of knowledge is "jacks or better to open".