We present the design of a frequency selective surface (FSS) with large harmonic rejection bandwidth. The FSS comprises five metal layers, each consisting of a periodic array of subwavelength elements. The design is based on an equivalent transmission line model that includes only the fundamental Floquet mode. This model is valid under the condition that the unit cell size is smaller than the distance between layers. The multi-layer FSS has an overall height of one sixth of the freespace wavelength at the central frequency of operation. The simulated frequency response exhibits good stability over a wide conical incidence range, up to 45 degrees. A prototype of the designed FSS is fabricated and combined with a wide-scanning connected array of dipoles to implement an antenna element for phased array applications, with integrated filtering properties.