The exponent in TropicalMatrixPower should be a positive integer, right? You are applying it to a matrix?
The first step in the iteration will give a zero matrix whenever the matrix a is positive:
Clear[a]
TropicalMatrixTimes[Array[a, {2, 2}], IdentityMatrix[2]]
To get a nonzero power, try a negative matrix a:
a = -{{2, 3}, {5, 7}};
b = {{4, 2}, {1, 2}};
TropicalMatrixPower[a, b]
