The canonical ladder operators for, say, orbital angular momentum are something like
$$ \hat L_+ = \hat L_x + i \hat L_y $$
and it can be shown that, if $ \left| \phi \right> $ is an eigenstate of $ \hat L_z $ with eigenvalue $m \hbar$ , then $ \hat L_+ \left| \phi \right> $ will also be an eigenstate with a new eigenvalue $ (1 + m)\hbar $ .
My question is how can we be sure that this is indeed the 'next' eigenstate, and we haven't missed one with an eigenvalue $ \lambda $ where $m < \lambda < 1+m $ ? How do we know that angular momentum is quantised in units of $ \hbar $ ?
In my limited experience, ladder operators are used to demonstrate the quantisation of angular momentum. I'm sure this is an oversimplification, and that these operators have been constructed and defined for this specific purpose, but is there an obvious way to see that they generate every eigenstate just from the form of the operators?