erislover, the mere fact that you are assuming an axiom does not imply that an argument is begging the question, unless it assumes the axiom. This should be obvious; how many times have you engaged in a discussion with them, only to abandon it because they refused to even consider alternatives? (“Okay, suppose for a second that God doesn’t exist.” “But he does!”) It’s irritating, isn’t it, to have somebody ignorantly declare your argument false (or irrelevant) merely because he has a preconcieved idea that he can’t ignore for even an instant? Well, you’re behaving in an irritating manner regarding this argument. Have you ever heard the phrase “for the sake of argument”? Well, for the sake of argument stop inserting extra premises into the argument and consider what is actually being said.
(And if you think that, in my DfD proof, Case 2’s “You don’t exist” and Case 1’s “You exist” are somehow contradictory, you need a refresher on proof by cases. I can write the proof in formal symbolic logic if you need me to. If you’re just jamming some spare “You exist” axiom into Case 2, then refer to previous paragraph.)
I’ve been thinking about this for a while, and after some reflection, I’ve decided I don’t see the problem. More accurately, I can’t tell which problem you are having.
Possibility 1: That you cannot accept “I am thinking” as a premise, because you think that you have to exist to think and you think that begs the question.
If this is your problem, then you misunderstood the structure of Descartes’ discussion. Descartes started by examining reality as we percieve it as an argument. That is, reality seems to imply certain things, and Descartes wondered wether, after doubting the assumptions that were already presented by reality, anything would be left. The thing is, in order to remove an axiom (such as “I have feet”), he had to add an axiom (“I doubt that I have feet”). Note that doubting something isn’t assuming the opposite; it merely means that you can no longer conclude things from the doubted premise.
After he got done annihilating most of reality, he was left something like the following:
“I exist”
“I have doubts”
and a big rack of doubts like “I doubt I have feet”, “I doubt I sense things”, “I doubt turnips”, etc.
He then went one step further.
“I doubt I exist”
“I have doubts”
and a big rack of doubts like “I doubt I have feet”, “I doubt I sense things”, “I doubt turnips”, etc.
From there, he pointed out that “I have doubts” implies “I exist”, and stops. Not exactly good form. I think philosopers are fundamentally afraid to consider that they don’t think, because if it’s true, then there’s no reason to keep feeding them. Let’s finish what he started.
“I doubt I exist”
“I doubt I have doubts”
and a big rack of doubts like “I doubt I have feet”, “I doubt I sense things”, “I doubt turnips”, etc.
So now all we have is a big rack of doubts. If you like, we can doubt that, leaving us with:
“I doubt I have a big rack of doubts.”
Much cleaner. Problem is, we can’t get rid of it entirely.
“I doubt I doubt I have a big rack of doubts.”
“I doubt I doubt I doubt I have a big rack of doubts.”
“I doubt I doubt I doubt I doubt I have a big rack of doubts.”
The longer you try to convince yourself that you have no axioms, the more your final axiom shouts that you are doubting things. It is not possible to rationally convince yourself that you are doubting by doubting the fact. Thus, the assertion “I doubt” remains long after you have doubted the assertion that you exist. And from that last remaining undoubtable assertion, you can conclude “I exist.”
By the definition provided by erislover, this argument isn’t circular, and arguments are only begging the question when the premise is as doubtful as the conclusion. Based on the additional reasoned fact that “I doubt” is undoubtable, we can conclude that Descartes’s argument is neither circular nor does it beg the question.
Possibility 2: Okay then, how do I know that that thing that is doing the doubting is me?
Well, for one, it sure seems like I’m the one having the doubts, so I seem to be in the defending camp. I can say “I doubt that I’m the one having the doubts” all day until my text font turns blue, but I undo the doubt each time I make it. 
That’s a semantic argument, you might protest, unamused. So what, I might reply; do you have a better definition of self than 'the one who is thinking my thoughts"? Remember, references to senses and memory are irrelevant by the time this becomes a problem. After you strip away all other identification, how else can you know yourself but as the owner of your thoughts? The ‘I’ isn’t the most important part of the argument anyway. After you prove that something exists, you’ve achieved your primary ontological goal.
Possibility 3: Hey, who knows? If the above two possibilities don’t tickle your fancy, feel free to explain your problem. Maybe I’ll understand you this time. Also, does my “Descartes for Dummies” proof clarify anything? I haven’t heard your take on it. In case you didn’t noticed it, It’s rather more to the point than Descartes himself was; plus it’s probably the least misinterpretable thing I’ve said yet on this thread! If you have any problems with the cogito, they ought to apply to that argument too, and be easier to point out there.