Well certainly you cannot obtain a transcendental number as a sum of algebraics since algebraics are closed under sums. Nor can you obtain every algebraic number as a sum of radical integers (else every polynomial would be solvable!). But it's easy to see that one can obtain any rational integer.
Note: In fact Heine originally defined algebraic integers to be radical integers, i.e. the ring obtained by closing $\rm\:\mathbb Z\:$ under taking $\rm\:n$'th roots. Heine claimed that every solvable algebraic integer is a radical integer - a problem which is still open according to Franz Lemmermeyer. However, it is not too difficult to show that every quadratic integer is a radical integer, e.g.
$ \frac{\sqrt{17}+1}{2}\ =\ \frac{\sqrt{17}+\sqrt{5}}{2}\ -\ \frac{\sqrt{5}-1}{2}\ =\:\ (7\ \sqrt{5} + 4\ \sqrt{17})^{1/3} - (\sqrt{5}-2)^{1/3}$