If you replace it with the cyclic shift operator, you get a circulant matrix (the same as your $L_n$ except that the bottom-left entry is $1$).  The eigenvalues of that matrix are the $n$th roots of unity.  So as $n$ grows, the spectrum fills the unit circle (it does not fill the unit disk, though).

Your $L_n$ is a highly non-normal matrix; the circulant version is normal.  If you want to understand this better, read Chapter 7 of Trefethen & Embree's *Spectra and Pseudospectra*, which deals specifically with your example.