The answer is probably no.
In his paper Transcendence of Periods: the State of the Art (Pure and Applied Mathematics Quarterly, Volume 2, Number 2, p. 435-463, 2006), Michel Waldschmidt conjectures that no period is a Liouville number (see questions 2 & 3 in the introduction). This expectation is supported by the main advances of transcendental number theory in 20th century, notably Baker's theory of linear forms in logarithms and its extensions to commutative algebraic groups.
Here, the word period has to be understood in the sense of Kontsevich and Zagier: the a complex number whose real and imaginary parts are values of absolutely convergent integral of an algebraic $d$-form rational functions with rational coeﬃcientson a , over domains in $d$-dimensional semi-algebraic set defined \mathbf R$ given by polynomials polynomial inequalities with rational coeﬃcients. (The preceding definition is quoted from the paper Periods and elementary real numbers by Masahiko Yoshinaga, who was apparently the first to prove that periods belong to the field of elementary complex numbers, those whose real and imaginary parts can be effectively approximated by Cauchy sequences of rationals.)