The non-linear dynamics of a series of salt-free surfactant solutions is investigated via a combination of TEM imaging, shear rheology and capillary breakup extensional rheometry. Surfactant solutions contain Hexadecyltrimethylammonium p-toluenesulfonate (CTAT) in de-ionized water. At low surfactant concentrations, 0.7 wt% < c < 1.1 wt%, solutions exhibit shear thickening behavior and the transient extensional measurements lead to formation of long lasting elastic filaments with a fairly constant extensional relaxation time lambda(E) = 0.03 s. At higher concentrations, 1.1 wt% <= c < 2 wt%, solutions exhibit strong viscoelastic behavior evidenced by dramatic increase in shear relaxation time. However, for concentrations above 2 wt%, the shear relaxation time decreases, while the zero shear viscosity reaches an asymptotic value. TEM images suggest that below the critical concentration, a polydisperse micellar structure with fairly long, short and spherical micelles co-exist. However, above the critical concentration, a dense micellar network forms with no indication of branched micelles. Using shear rheology data, we demonstrate that the micellar chain length decreases beyond the critical concentration, while the entanglement number remains fairly constant. On the other hand, the maximum Trouton ratio measured in transient extensional experiments decreases as surfactant concentration is increased to c approximate to 3 wt%, and remains fairly constant around Tr approximate to 3, for higher concentrations. This finding is in-line with the structural parameters inferred from the shear rheology and also the TEM images. Therefore, we confirm that the extensional rheology is sensitive to structural transitions near the critical concentration. Finally, we show that the ratio of the extensional relaxation time to the shear relaxation time, lambda(E)/lambda, is always less than unity regardless of whether relaxation is dominated by reptation or breakage of the wormlike micelles.