Digital filters with more zeros than poles
If you consider the transfer function of a causal IIR filter

$$H(z)=\frac{B(z)}{A(z)}=\frac{\sum_{m=0}^M b_mz^{-m}}{\sum_{n=0}^N a_nz^{-n}},\quad a_0=1$$

then you always get the same number of poles and zeros, regardless of the choice of $M$ and $N$ (as already pointed out by Robert). However, what is meant by a system with "more zeros than poles", is a system with a numerator degree $M$ greater than the degree $N$ of the denominator. In this case "poles" refers to the poles away from the origin. It is only these poles that have to be implemented. So a system with some poles at the origin is cheaper to implement than a system with all its poles away from the origin. So your argument that you "might as well use those poles ... if you have them" does not really hold, because poles at the origin come without any implementation costs.

The remaining question is why such filters with more zeros than poles (away from the origin) can be useful. Let me just give two examples:


In implementation as well as in the design process, poles can give a lot of trouble (instability, noise enhancement, local optima in the design process, etc.). On the other hand, FIR filters with good frequency selective properties must often have a very large degree (resulting in a large delay and in high implementation costs). Adding just a few poles can give us the best of both worlds: better filter behavior at a greatly reduced (numerator) degree $M$, and only little trouble because there are only very few poles.
For frequency selective filters with a small desired phase distortion in the passbands, IIR systems with $M&gt;N$ are very useful, because the poles ideally contribute only to the passbands, whereas the zeros must contribute to the passbands (for phase equalization) as well as to the stopbands (zeros on or close to the unit circle). Consequently, we need poles only at angles corresponding to passband frequencies, but we need zeros everywhere.


The figure below shows such a system (lowpass filter with approximately linear passband phase response, $M=12, N=6$):