Is it possible to write an infinite sentence that is grammatically correct?

The answer is no. A sentence needs punctuation. After that punctuation it is over. A sentence fragment can be infinite but not a whole sentence because it has to end.

According to Wikipedia, such a sentence is theoretically possible.

In practice, of course, though the clauses and other structures of even the longest sentence may be notionally separated by such typographical devices as semi-colons, dashes, parentheses and so, in practice that sentence will consist of shorter sentences which might as well be marked off at intervals by full stops/periods as by the various pseudo-periods I've just mentioned.

There is no reason that a computer couldn't be coded to write a recursive sentence along the lines of

I wrote down a 1, and then I wrote down a 2, and then I wrote down a 3, and then I wrote down a 4, ...

with a simple k + 1 increase, indefinitely extended. Of course, a machine wouldn't reach infinity with that operation, any more than computers calculating π or simply counting integers ever would.

But more to the point, it's hard to argue that such a sentence has any theoretical interest, since it amounts to a snippet of language infinitely repeated with one minor and predictable variation in each iteration. Is it grammatically correct? Sure. Is it coherent? Enough so that no one would be tempted to spend more than an extremely finite period of time reading it. But so what?

The problem with the most obvious infinitely long sentences is that their trajectory is too boringly easy to comprehend, not that they are overwhelmingly complicated. And the direction of the simple case is enough to persuade me that the undertaking itself isn't very interesting.

There is no theoretical limit on the length or depth of a sentence. But

• An infinite sentence cannot be uttered, because the utterer will die before it can be completely encoded, and

• An infinite sentence cannot be understood, because it cannot definitively decoded until it reaches its end; but a sentence which reaches its end is not infinite.