There are an infinite number of irrational numbers between any pair of numbers you care to name. Some are "algebraic" in that they are roots of polynomials. Some are not. You can make up one simply by identifying a rule that can be used to determine the value.
π + 2.2 ≈ 5.34 . . . . sum of rational and irrational
√29 ≈ 5.39 . . . . . . root of a number that is not a square
5.21234567891011121314151617181920... digits are sequential integers
≈ 5.21
∛150 ≈ 5.31 . . . . . . root of a number that is not a cube
