It is always difficult, in most cases impossible, to answer the question of the type "who was the first", especially who was the first to discover some trivial thing, which everyone can see once s/he looks at it. By "trivial" in this case I mean that once you start considering iterations
of a continuous function, you immediately discover that they can converge only to a fixed point. Thus a more reasonable question would be "who was the first to consider infinite iteration of continuous functions". The earliest instance that I know is Euler's paper
De formulis exponentialibus replicatis. Acta Acad. Petropolitanae 1 (1777), 38–60.
where he considers the existence of the limit
This can be easily reformulated as iteration of exponential function.
Euler himself says in the paper that he follows the remarkable work of Marquis de Condorcet. But the paper of Condorcet does not address the question of convergence, he just iterates some formal expressions.
But of course I cannot prove that "Euler was the first", and whatever other answer to this question is given, it can never be proved. That's why I think this is a meaningless question, as all questions of this type are.
Remark 1. Of course, Babylonians had an algorithm of extracting the square root,
which is equivalent to the Newton method for the equation $x^2-a=0$.
But to say that Babylonians were considering "convergence to a fixed point of iterations of a continuous function" would be a ridiculous anachronism.
Euler at least understood the words "function", "limit" and "fixed point" more or less as
we understand them now, though not exactly in the same way.
Perhaps it is interesting to mention that Euler found when the limit exists for real $x$. In the subsequent centuries there was a great interest to these problems in complex domain, and complete understanding of possible limits was achieved only in 2009 (J. Hubbard, D. Schleicher and M. Shishikura
Journal: J. Amer. Math. Soc. 22 (2009), 77-117).