The prefix palindromic length PPL_u(n) of an infinite word u is the minimal number of concatenated palindromes needed to express the prefix of length n of u. In a 2013 paper with Puzynina and Zamboni we stated the conjecture that PPL_u(n) is unbounded for every infinite word u that is not ultimately periodic. Up to now, the conjecture has been proven for almost all words, including all words avoiding some power p. However, even in that simple case the existing upper bound for the minimal number n such that PPL_u(n) > K is greater than any constant to the power K. Precise values of PPL_u(n) are not known even for simplest examples like the Fibonacci word.

In this paper, we give the first example of such a precise computation and compute the function of the prefix palindromic length of the Thue-Morse word, a famous test object for all functions on infinite words. It happens that this sequence is 2-regular, which raises the question if this fact can be generalized to all automatic sequences.