A differential equation $y'=F(y)$ may be called stiff in $[0,T]$ if, for some $y$, the matrix $TF'(y)$ has some huge eigenvalues with negative real part.
Thus the simplest example is $y'=-y$.
Indeed, the equation $y'=-10^6 y$ for $t\in[0,1]$ is obviously stiff, but $y'=-y$ is as stiff for $t\in[0,10^6]$. An explicit method such as Euler's needs the same number of integration steps to achieve the same accuracy.
Whether the example is interesting depends on your interests. But by the same argument, all dissipative ODEs become stiff when integrated over huge time intervals. As all ODEs are (by definition) dissipative if the trajectory approaches a lower-dimensional stable manifold, there is no paucity of realistic examples.