I want a reference to the literature of a power series convergent in the whole CLOSED unit disk,but unbounded there.
An example of such a series is given in:
W. Sierpiński "Sur une série potentielle qui, étant convergente en tout point de son cercle de convergence, représente sur ce cercle une fonction discontinue" Rendiconti del Circolo Matematico di Palermo, Volume 41, Issue 1, pp. 187-190
I learned about this from Julien Melleray in an answer to the following question Does a power series converging everywhere on its circle of convergence define a continuous function? (which seems very close but then not the same as this one).
Also the answer of George Lowther there is very interesting.
To avoid potential confusion let me stress that while the title of the paper talks about discontinous, the function constructed by Sierpiński is actually unbounded (and thus discontinous).