The simplest non-Kekulé molecules are biradicals. A biradical is an even-electron chemical compound with two free radical centres which act independently of each other. They should not be confused with the more general class of diradicals.
One of the first biradicals was synthesized by Wilhelm Schlenk in 1915 following the same methodology as Moses Gomberg's triphenylmethyl radical.  The so-called Schlenk-Brauns hydrocarbons are:
Eugene Müller, with the aid of a Gouy balance, established for the first time that these compounds are paramagnetic with a triplet ground state.
Another classic biradical was synthesised by Aleksei Chichibabin in 1907.  Other classical examples are the biradicals described by Yang in 1960 and by Coppinger in 1962.