Transcendental numbers are any numbers that cannot be expressed as a root of a polynomial equation.  Their most visible feature is that they do not repeat; you cannot write the last digit that specifies the number completely. Some of the popular transcendental numbers include  "π" and "e".  
Most real and complex numbers are transcendental, since the algebraic numbers are countable while the sets of real and complex numbers are both uncountable. All real transcendental numbers are irrational, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental.

vCalc's Transcendental library contain such kind of numbers.