In his Unbelievable discussion, Max Andrews stated that he has noted in himself, when shaping apologetic arguments, the tendency to beg the question of god. You may recall that begging the question is, at base, a fallacy of circular reasoning, even though the circularity might be subtle. It appears that he and I might have some alignment here in that I have noticed, on the part of apologists, the exact same tendency. That said, while I agree that many arguments for god presume god, or something attributed to god, I would not, however, have necessarily included fine-tuning based arguments in the list.
These are the apologetic arguments that, off the top of my head, include some element of begging the question:
Argument from First Cause
Argument from Resurrection
Argument from Revelation
Argument via Inference to Best Explanation
A good example of the sort of apologetic question begging involved appears in this video by Dave S at theglobalatheism, which concludes that the ontological argument proves no more than that if god exists, then god exists (which is the law of identity: if X, then X, or X=X).
Another video by Dave S, on the argument from resurrection, also hints at begging the question, pointing out that to get a high-ish likelihood that god exists (Pr≈0.5) you need to assume that claims that god raised people from the dead were true. Think about this for a moment. If you assume, up front, that claims that your god raised people from the dead are true, this necessitates the assumption that your god actually exists. You should find this a little odd, because it would mean that, rather than adding support to the notion that the apologist's god exists, the resurrection argument reduces the likelihood of that god's existence. This seems odd to me.
Anyway, what about the fine-tuning argument. Does this include an element of begging the question?
Perhaps there is. The fine-tuning argument first suggests that the universe is fine-tuned (arguing the reality of fine-tuning is a speciality of Andrews' partner in the discussion, Luke Barnes). But that's insufficient to show god. To show god, the apologist has to go one step further and argue that nothing else explains fine-tuning other than their god.
This is where a question can be comprehensively begged, on two levels.
Firstly, there is the definition of "god". As indicated above, it can be argued that the ontological argument can fail based on the fact that it tries to define a god into existence. The fine-tuning argument fails on the same grounds, because of begging the question. The apologist pressing this argument is effectively trying to define their god into existence via the claim that their god can do anything that is necessary to prove that existence. Their god can make a universe that looks like ours and seems incredibly unlikely, but you can bet your bottom dollar that if it ever starts to look like the universe is incredibly likely, the same apologist will argue that their god can make an incredibly likely universe as well. Or an eternally inflating multiverse. Or a quantum multiverse. Therefore, their argument collapses into "if our universe were created by the sort of being that can make universes of the sort that we live in, then our universe was created by the sort of being that can make universes of the sort that we live in, let's call it god".
Secondly, there is the sheer unlikeliness of our universe versus typicality. In this argument, there is no assumption of a god followed by a conclusion of a god, it's more an assumption of atypicality followed by a conclusion of atypicality which feeds into a "this is so surprising, therefore god" argument. Ignoring for a moment the fact that multiverse theory arises totally independently of counter-apologetics, if we were to posit a multiverse to solve the unlikeliness of our universe, the counter-counter-apologists then rush in to say that, if there were many different kinds of universes in this multiverse, then our type of universe would be atypical - and, therefore, we have an obligation to explain why we are in this atypical universe.
This atypicality is often expressed in terms of a Boltzmann multiverse, the sort that fails on the basis of Boltzmann brains. In a Boltzmann multiverse, each universe is a fluctuation of order in a vast, enduring region of comprehensive disorder. In such a regime, small regions of order would be far more likely than regions like our observable universe and we would indeed be unusual. However, all this does is put a kibosh on a Boltzmann multiverse. But no-one is seriously positing a Boltzmann multiverse (I will do so semi-seriously in a later article though).
Nevertheless, when people regurgitate WLC's argument against the multiverse (including people like Barnes) they assume a Boltzmann multiverse and conclude that a multiverse is impossible. Thus they are arguing that an impossible form of multiverse is impossible, which is trivially true.
The other element of the argument is based on the assumption that we must be typical. This seems similar to the argument that leads to (counter-apologetically derived) multiverses, but there is a subtle but important difference between very highly unlikely and very highly atypical. When you bring the multiverse to bear on the problem, the likelihood of a very highly unlikely universe existing increases, towards 100% as N (the number of universes) approaches infinity. At the same time, as N approaches infinity, the number of atypical universes increases, the number of distinct chances of being atypical increases.
Think of it in terms of jellybeans with the probabilities being indicative, rather than rigorous. If one in every 100 jellybeans was atypical, then with a sample of 100 jellybeans, you would expect on average that one would be atypical and there would be only chance to be atypical. If there were 10,000 jellybeans, then we'd expect there to be 100 chances to be atypical. Agreed, there would still be a lot more typical jellybeans than atypical ones, but the increased number of atypical jellybeans would increase commensurate with sample size. If we introduced a filter that only permits a certain shape of atypical jellybean to pass through, and used a very large, approaching infinite sample size, then so long as the shape is possible, then we should not be surprised to see one.
This is the situation that we find ourselves in. We are in a universe which, although atypical, must be the way it is for us to find ourselves in it - because it's precisely the sort of universe in which we could develop as persistent intelligent life (rather than Boltzmann brains).
In other words, our atypicality is very much the point, rather than being a failure of the multiverse to explain our fortuitous existence.
Curiously enough, Dave S includes, as further reading for the resurrection video, which was posted a year ago, both my Sweet Probability and Max Andrews' dissertation on Bayes Theorem and the resurrection. Some nice recognition there for both of us, even though I at least hadn't noticed it.