  Laws of the iterated logarithm of "limsup" type are studied for
multi-dimensional selfsimilar processes $\{X(t)\}$ with independent increments
having exponent $H$. It is proved that, for any positive increasing function
$g(t)$ with $\displaystyle \lim_{t\to\infty}g(t) = \infty$, there is $C\in
[0,\infty]$ such that $\limsup|X(t)|/(t^Hg(|\log t|))= C$ a.s. as $t \to\infty
$, in addition, as $t \to 0$. A necessary and sufficient condition for the
existence of $g(t)$ with C=1 is obtained. In the case where $g(t)$ with C=1
does not exist, a criterion to classify functions $g(t)$ according to C=0 or
$C=\infty$ is given. Moreover, various "limsup" type laws with identification
of the positive constants $C$ are explicitly presented in several propositions
and examples. The problems that exchange the roles of $\{X(t)\}$ and $g(t)$ are
also discussed.
