The Irrationality Exponents of Computable Numbers
  We prove that a real number a greater than or equal to 2 is the irrationality exponent of some computable real number if and only if a is the upper limit of a computable sequence of rational numbers. Thus, there are computable real numbers whose irrationality exponent is not computable. 
Ver\'onica Becher and Yann Bugeaud and Theodore A. Slaman
math.NT