We report on the optical properties of single isolated silver nanodisks and pairs of disks fabricated by electron beam lithography. By systematically varying the disk size and surface separation and recording elastic scattering spectra in different polarization configurations, we found evidence for extremely strong interparticle interactions. The dipolar surface plasmon resonance for polarization parallel to the dimer axis exhibited a red shift as the interdimer separation was decreased; as expected from previous work, an extremely strong shift was observed. The scattering spectra of single particles and pairs separated by more than one particle radius can be well described by the coupled dipole approximation (CDA), where the particles are approximated as point dipoles using a modified dipole polarizability for oblate spheroids. For smaller particle separations (d < 20 nm), the simple dipole model severely underestimates the particle interaction, indicating the importance of multipolar fields and finite-size effects. The discrete dipole approximation (DDA), which is a finite-element method, describes the experimental results well even at d < 20 nm, including particles that have metallic bridges.