X-ray, UV and optical spectra are presented for a rare, EUV-bright Seyfert galaxy, RE J1034+396 (z=0.042). When combined, these data describe a very-high temperature big blue bump (BBB; kT_eff~100 eV) whose high energy turnover is observed in soft X-rays. The soft X-ray/UV flux ratio is extremely high, L_0.2keV/L_1200A~10, compared to AGN in general whose ratio is usually less than 1. The optical/UV continuum betrays no trace of the BBB or the host galaxy. It rises steeply to the red, and is consistent with an underlying far-IR to X-ray power law with an index of ~1.3. This may represent the first direct identification of an optical power-law component in a non-blazar AGN. We compare the optical to soft X-ray spectrum of RE J1034+396 with three mechanisms for the production of the BBB: an optically thick accretion disc (AD); optically thin emission from a hot plasma; and reprocessing in cool clouds. A thin AD can reproduce the observed spectrum of RE J1034+396, but a free-free spectrum predicts too much flux in the UV. The reprocessing of a non-thermal continuum is only possible if there is a steep break between ~60 and 1000A a reprocessed free-free spectrum is an alternative. We compare the data with those of the high-redshift quasar E1346+266, whose BBB component is similar in shape, but 200 times stronger than that of RE J1034+396. A simple, thin AD is not appropriate for E1346+266 because of its high luminosity, but, when electron scattering is taken into account, an adequate representation is obtained for a black hole mass M=2.3x10^8 M_solar with an accretion rate of 2.7 M_solar yr^-1 (L/L_Edd~2). Optically thin (T_brem=7x10^6K) emission and reprocessed free-free are also possible for E1346+266.