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 integral of an algebraic $d$-form with rational coefficients on a $d$-dimensional semi-algebraic set defined by polynomials with rational coefficients.