RootApproximant does a very good job when I need to recognize an algebraic number when enough its digits are known (or even when an unlimited number of digits can be obtained from a numerical computation given enough time).

But often I need to recognize linear combinations of the form $\alpha+\beta\cdot\tau$ where $\alpha$ and $\beta$ are unknown algebraic numbers, and $\tau$ is a known fixed transcendental number (e.g. Pi, E or Log[2]).

Some simple cases can be solved using WolframAlpha lookup:

WolframAlpha["2.421441469079183123",
    IncludePods -> "PossibleClosedForm",
    TimeConstraint -> ∞]

Is there a way to solve this problem in general in Mathematica?