Can you cast Gentle Repose multiple times in order to extend the "revivability" of a body as much as you like?

Let's take a look at the text of the gentle repose spell:

Casting Time: 1 action
Range: Touch
Components: V, S, M (a pinch of salt and one copper piece placed on each of the corpse's eyes, which must remain there for the duration)
Duration: 10 days

You touch a corpse or other remains. For the duration, the target is protected from decay and can't become undead.
The spell also effectively extends the time limit on raising the target from the dead, since days spent under the influence of this spell don't count against the time limit of spells such as raise dead.

p. 245 PHB

The answer is yes.

As long as you are able to cast the spell again before the duration ends, you are effectively stopping the clock for the 10-day window of the reincarnate spell. Also, take note there is a stipulation in the spell components that the two copper pieces must remain on the corpse's eyes for the duration.

Does this make a character unkillable?

The gentle repose spell does not have an effect on living characters, so this would, in fact, do nothing to make a character "unkillable."

Keep in mind that casting the reincarnate spell on a protected corpse does not guarantee they will return to life:

Provided that the creature has been dead no longer than 10 days, the spell forms a new adult body for it and then calls the soul to enter that body. If the target's soul isn't free or willing to do so, the spell fails.

p. 271 PHB (emphasis mine)

A creature's soul could be prevented from returning to life for a number of reasons in the time between its death and attempted resurrection.

He cannot cast the spell if the conditions for its casting aren't met.

This is distinct from him casting the spell and the spell failing, mind you. In order to cast the spell, he must be within the range specified by the spell, which in this case is touch. Now there is an exception to this in this spell, which applies if the body no longer exists. Given that the body is still in existence, I would say that the spell fails should he decide to try and cast it regardless but that the spell slot is not consumed. The reasoning behind this is that the range of the spell is part of its requirements for its casting, as much as any component, so casting True Resurrection without being in range would have the same result as casting it without a diamond: the process of casting it remains incomplete.

That said, if you are using the optional rule on Xanathar's Guide to Everything (p. 86), under "Invalid Spell Targets", the spell slot would be expended:

If you cast a spell on someone or something that can't be affected by the spell, nothing happens to that target, but if you used a spell slot to cast the spell, the slot is still expended.

That said, any high-level cleric would be aware of this, and would most likely not cast the spell.