You cannot fully characterise a general state without measuring a full basis of operators. To characterise an $m$ dimensional $\rho$ you need $m^2-1$ (independent) measurement outcomes. Measure anything less than that, and you won't know the projection of the state on some axis (in general).
If say you only measure $XX,YY,ZZ$ on a two-qubit state, you won't be able to know what's $\langle XY\rangle$ etc. This might change if there are underlying assumptions on the state.
It is true that measuring $XXX,YYY,ZZZ$ is sufficient to fully characterise your state if you assume that it is of "GHZ-class". By which I mean, if the state is $|\psi_{a,b}\rangle= a|000\rangle+b|111\rangle$, then
$$\langle XXX\rangle = 2ab,
\qquad \langle YYY\rangle = 0,
\qquad \langle ZZZ\rangle = a^2-b^2,$$
which means you can retrieve $a,b$ from the measurement results (and in fact only $XXX$ and $ZZZ$ are needed). However, this is only because you made a (quite strong) assumption on the form of the input state.
If you consider different types of maximally entangled states, say, $W$ states, then this already stops being true.
Note for example how all states of the form
$$a|001\rangle + b|010\rangle + c|100\rangle$$
give $\langle XXX\rangle=\langle YYY\rangle=0$ and $\langle ZZZ\rangle=-1$, regardless of the values of the coefficients $a,b,c$.