That's a subtle question because it depends on what you mean by energy.
Suppose we take some object a long way from the black hole and let it fall. As it falls it's speed increases (obviously) and therefore it's kinetic energy increases. However if you are sat outside the black hole watching the object you'll see something strange as the object nears the event horizon. You'll see the object start to slow as it approaches the horizon and indeed at the horizon it will come to a complete stop. This is the source of the numerous questions asking whether anything can really fall into a black hole e.g. this and this. Incidentally while searching for such questions I found the question A Conflict with Black Holes that is relevant to your question.
So from our perspective outside the black hole the kinetic energy of the body rises to a maximum then falls again as it nears the horizon.
However there is more to this. Suppose you jump on a powerful rocket and hover at some distance $d$ from the event horizon, then you measure the velocity at which the object passes you. As you approach the horizon, i.e. as $d \rightarrow 0$, the velocity the object passes you will tend to $c$ and therefore it's energy will tend to infinity. However you cannot hover at the event horizon (that would take an infinitely powerful rocket) so no observer will ever see the velocity become $c$ and the kinetic energy become infinite.
The other option you might take is to jump into the black hole alongside the object so you are falling with it. This is actually the least interesting option (until you hit the singularity and the associated spectacularly messy death of course) because you will simply see the object float, weightless, beside you and you won't measure any change in it at all.