Edward, the proprietor of Beyond Necessity, presents an infinite regress argument against truth-makers. Here it is:
. . . I reject the idea of a truthmaker altogether. If there is such a truthmaker, let it be A, it comes into existence when Socrates sits down, and ceases to exist when he stands up. If it were something real – let’s say a candle flame, which comes into existence when we light the candle, and ceases to exist when we blow it out – then there would have to be a further truthmaker for A existing. I.e. the sentence “A exists” can be true or false, and so requires a further truthmaker B, that makes it true when B exists. But then “B exists” requires yet another truthmaker, and so on ad infinitum. That is absurd. Therefore, there are no truthmakers.
I am not sure Ed understands what a truth-maker is. Here is a Philosophy 101 explanation. Suppose we have some true contingent declarative sentence such as 'Tom is tired.' The truth-maker theorist maintains that for contingent true sentences, there is more to the sentence than its being true. There must be something external to the sentence, something that is not a sentence, that 'makes it true.' If you deny this, then you are saying that the sentence is just true and that there is no explanation of its being true in terms of anything extralinguistic. And surely that is absurd, assuming you are not some sort of linguistic idealist. 'Tom is tired' cannot just be true; it is true because there exists a man to whom 'Tom' refers and this man is in a certain state.
Could Tom by himself be the truth-maker of 'Tom is tired'? No. For if he were, then he would also be the truth-maker of 'Tom is manic' -- which is absurd. This is why truth-maker theorists (not all but most) introduce facts or states of affairs as truth-makers. David Armstrong is a prominent contemporary example.
Now what are we to make of Edward's argument? The argument seems to be that if sentence s has a truthmaker t, then the sentence 't exists' must also have a truth-maker, call it t*. But then the sentence 't* exists' must itself have a truth-maker, t**, and so on ad infinitum.
Now this is a terrible, a thoroughly and breath-takingly rotten, argument which is why no one in the literature (to the best of my knowledge) has ever made it. Suppose that 'Tom is tired' is made-true by the fact of Tom's being tired. Call this fact F. If 'Tom is tired' is true, then F exists, whence it follows that 'F exists' is true. (This of course assumes that there is the sentence 'F exists,' an assumption I will grant arguendo.) Since 'F exists' is contingent, we can apply the truth-maker principle and ask for its truth-maker. But surely its truth-maker is just F. So there is no regress at all, let alone an infinite regress, let alone a vicious infinite regress. (Please note that only vicious infinite regresses have the force of refutations.) 'Tom is tired' has F as its truth-maker, and 'F exists' has the very same F as its truth-maker. Tom's being tired makes true both 'Tom is tired' and 'Tom's being tired exists.' No regress.
So Ed's argument is a complete non-starter. There are, however, plausible arguments against facts as truth-makers. See my Facts category.