The question was apparently first raised by Brown and Freedman in a 1987 paper, then independently by Pirillo and Varricchio in a 1994 paper, and by Halbeisen and Hungerbühler in 2000.
It follows from results of Dekking that such a word exists avoiding four consecutive blocks.
Recent results of Cassaigne, Currie, Schaeffer, and Shallit (2011) show that such a word exists avoiding three consecutive blocks.
-- JeffreyShallit - 13 Jul 2011