The realisation that the atom was divisible, and thus that elements were not the "fundamental building blocks" arguably began around 1897 with J. J. Thomson's discovery of the electron and proceeded through the early 20C to Rutherford's experiments with hydrogen nuclei, described as the discovery of the proton (although William Prout [ibid] had suggested that all atoms were composed of hydrogen atoms as early as 1815).
From this Wiki page, Robert Boyle proposed the idea that matter was made of irreducible atoms in 1661, rejecting earlier ideas of "the elements", and in 1789, Antoine Lavoisier published a list of 33 elements (although some were questionable). By 1818, atomic weights were known for 45 of the 49 accepted elements; in 1869, Mendeleev had 66 elements in his periodic table; and by 1900, all but three of the naturally occurring stable elements had been discovered.
So the question is: as the number of known elements grew (see graph here), did anyone (ideally with at least some degree of credibility) use what might be termed a more "philosophic" argument to question the fundamental nature of elements and atoms, based on the idea that there were "too many" elements for them to be "fundamental"?
The "thought process" (not necessarily "logical objection") that I was wondering whether anyone had gone through was along the lines of:
"So, we've now got 50 [or however many] fundamental particles, with more being found every year... maybe they are not quite so fundamental as we thought... maybe there is something else [protons / neutrons / electrons in the first descent] going-on underneath?"
There is, I suppose, no intrinsic reason why there couldn't be a large / infinite number of "fundamental particles", and possibly such a situation only "feels wrong" to me because of the knowledge that atoms aren't fundamental. In essence, the question is about whether anyone else "felt this was wrong" before it was known that atoms were not fundamental. I'm suspecting the answer is "no".