DFT of discrete signals, why do we only analyze frequency bins equal to number of input samples?
A DFT is an invertable (square matrix) transform, thus the complex result vector is the same size as the complex input vector.  But you can analyze as many (additional) frequencies as you want by interpolating the results after the DFT (which requires additional computation that doesn't really increase separation resolution, but can produce a smoother looking plot).

You can analyze less if you want by using the Goertzel algorithm, which can be used to compute any single frequency result bin of a DFT.  (But calculating less than N orthogonal bins won't be invertable or "complete".  And any more than log(N) bins will likely be a lot slower than just doing a full FFT and throwing away what you don't need.)