Check if number is irrational/transcendental

istranscendental[x_] := ! Element[x, Algebraics]

To the extent that Mathematica is aware of the algebraic numbers, this should work. EulerGamma, for example, is returned unevaluated, while Pi returns True and 2 returns False.

We can use RootApproximant and PossibleZeroQ to guess if a number is algebraic or not.

PossibleAlgebraic[x_] := 
  With[{res = Element[x, Algebraics]},
    res /; BooleanQ[res]
  ]

PossibleAlgebraic[x_?NumericQ] /; !InexactNumberQ[x] := 
  With[{guess = RootApproximant[x]},
    Quiet[PossibleZeroQ[x - guess]] /; Element[guess, Algebraics]
  ]

PossibleAlgebraic[_?NumericQ] = False;

Some tests:

PossibleAlgebraic[Sqrt[2]]

True

PossibleAlgebraic[HypergeometricPFQ[{1/5, 2/5, 3/5, 4/5}, {1/2, 3/4, 5/4}, 3125/256]]

True

PossibleAlgebraic[π]

False

PossibleAlgebraic[EulerGamma]

False