We study the behavior of max-algebraic powers of a reducible nonnegative matrix
A
∈
R
+
n
×
n
A\in \mathbb {R}_+^{n\times n}
. We show that for
t
≥
3
n
2
t\geq 3n^2
, the powers
A
t
A^t
can be expanded in max-algebraic sums of terms of the form
C
S
t
R
CS^tR
, where
C
C
and
R
R
are extracted from columns and rows of certain Kleene stars, and
S
S
is diagonally similar to a Boolean matrix. We study the properties of individual terms and show that all terms, for a given
t
≥
3
n
2
t\geq 3n^2
, can be found in
O
(
n
4
log
n
)
O(n^4\log n)
operations. We show that the powers have a well-defined ultimate behavior, where certain terms are totally or partially suppressed, thus leading to ultimate
C
S
t
R
CS^tR
terms and the corresponding ultimate expansion. We apply this expansion to the question whether
{
A
t
y
,
t
≥
0
}
\{A^ty,\; t\geq 0\}
is ultimately linear periodic for each starting vector
y
y
, showing that this question can also be answered in
O
(
n
4
log
n
)
O(n^4\log n)
time. We give examples illustrating our main results.