The quadratic and the symmetric L-groups are 4-fold periodic.
What is the simple argument to obtain the intuition behind the 4-fold periodicity of $L$-theory?
(For example, why not have the Bott periodicity, say of the 2-fold and the 8-fold periodicity? as a comparison.)
To give two not so deep reasons:
The L-theory groups are defined so that the middle dimensional homology of a stably framed manifold have some type of quadratic structure. For simplicity, let us assume we have such an n-manifold M which has trivial homology up to the middle dimension. If $n=2k$ then $H_k(M)$ has the structure of a $(-1)^k$-quadratic form, where the underlying symmetric form is coming from Poincare duality, and the quadratic refinement is coming from counting some type of self intersections of spheres that represent the homology classes.
If $n=2k+1$, the pair $H_k(M)$ and $H_{k+1}(M)$ have the structure of a $(-1)^k$ quadratic formation (see Ranicki's "Algebraic and Geometric Surgery"). The L-theory groups are certain quotients of the collections of all these types of forms, and the middle dimensional homology considered inside these groups is a complete obstruction to doing framed surgery to make $M$ a sphere.
Quinn came up with the L-theory space directly in terms of these surgery questions, and so we see that at the very least the repetition in the homotopy groups (which are the surgery obstruction groups) is coming from the fact that the middle dimensional homology has 4 different structures depending on dimension.
The second reason (I suspect) is coming from the fact that if we invert 2, $BO$ has 4 fold periodicity. In the work of Kervaire and Milnor, they show that in these surgery questions 2-torsion is easily dealt with. So if we elect to invert 2, questions about vector bundles (which are ever present throughout surgery theory) will be answered with a 4-fold periodicity. I suspect one will find very complete ideas along these lines inside the book "The Classifying Spaces for Surgery and Cobordism of Manifolds" by Madsen and Milgram. There are many occurences of 4-fold periodicity in this book, most often arising in studying the spaces of normal invariants: $G/O, G/PL, G/Top$. These normal invariants sit in an exact sequence with the structure set and the surgery obstruction groups of a manifold.