Context. The radio quasar 3C 454.3 underwent an exceptional optical outburst lasting more than 1 year and culminating in spring 2005. The maximum brightness detected was R = 12.0, which represents the most luminous quasar state thus far observed (MB ∼ -31.4). Aims. In order to follow the emission behaviour of the source in detail, a large multiwavelength campaign was organized by the Whole Earth Blazar Telescope (WEBT). Methods. Continuous optical, near-IR and radio monitoring was performed in several bands. ToO pointings by the Chandra and INTEGRAL satellites provided additional information at high energies in May 2005. Results. The historical radio and optical light curves show different behaviours. Until about 2001.0 only moderate variability was present in the optical regime, while prominent and long-lasting radio outbursts were visible at the various radio frequencies, with higher-frequency variations preceding the lower-frequency ones. After that date, the optical activity increased and the radio flux is less variable. This suggests that the optical and radio emissions come from two separate and misaligned jet regions, with the inner optical one acquiring a smaller viewing angle during the 2004-2005 outburst. Moreover, the colour-index behaviour (generally redder-when-brighter) during the outburst suggests the presence of a luminous accretion disc. A huge mm outburst followed the optical one, peaking in June-July 2005. The high-frequency (37-43 GHz) radio flux started to increase in early 2005 and reached a maximum at the end of our observing period (end of September 2005). VLBA observations at 43 GHz during the summer confirm the brightening of the radio core and show an increasing polarization. An exceptionally bright X-ray state was detected in May 2005, corresponding to the rising mm flux and suggesting an inverse-Compton nature of the hard X-ray spectrum. Conclusions. A further multifrequency monitoring effort is needed to follow the next phases of this unprecedented event.