Examples in which the size of the alphabet ($\geq 2$) used for an encoding matters
Dinur's proof of the PCP theorem relies heavily on the manipulation of the alphabet size. 

Specifically, the overall structure of the proof is an iterative application of a graph powering technique a logarithm in the graph size number of times. On each iteration, the graph is pre-processed into a regular expanding graph, amplified by a power (which blows up the alphabet size), and then a PCP composition is applied (turning each constraint over a large alphabet into a system of constraints over a small alphabet).

The implicit goal of the process is to find a way to re-use the amplification step until the UNSAT value becomes a constant fraction (proving the PCP theorem). The key point is that unless the alphabet size is pulled back each time, the resulting graph is not what is needed for the final reduction.