Answer
Momentum and kinetic energy do not have any maximum value.
Work Step by Step
We can write an expression for the relativistic mass of an object when it is moving at speed $v$:
$m_{rel} = \frac{m}{\sqrt{1-\frac{v^2}{c^2}}}$
We can write an expression for the momentum:
$p = m_{rel}~v$
We can write an expression for the kinetic energy:
$K = \frac{1}{2}~m_{rel}~v^2$
We can see that as $v$ gets closer to $c$, the value of $m_{rel}$ increases without any upper limit. Therefore, momentum and kinetic energy do not have any maximum value.
