Getting PhD from average institution (gain experience) before applying for a PhD in top institution? What do you think about an idea of getting a PhD in an "better then average institution" to get tons of experience to get a second PhD (in the same field) from top institution?

I'm now getting a MSc degree in applied math from one of the top Russia institutions, but I still feel like I don't have enough experience, and publications, and general "coolness" for applying to places like MIT or Stanford or other top universities - but I would really want to work with people there! Moreover the problem I've came up with for my thesis is somewhat "non-mainstream" (that's why I don't really have a thesis adviser and I'm guiding myself through it, so it turned out to be slower process than I expected).

I've been thinking of getting a first PhD in some good place in Europe, getting few publications, gaining more experience there and only after that applying into top institutions. From your own experience, does that seem to be a good idea? (I mean, having a completed PhD in some other institution seems like a great advantage from the lab's perspective - right?)
Terrible idea, at least as far as a life goal is concerned. You will spend 5 years of your life doing something you consider only preparatory, without any guarantee that you will actually be admitted by the MITs and Stanfords of your aspiration. In fact, if we had someone applying to our graduate program who already has a PhD, we'd seriously be asking ourselves why this person wants to get another PhD? So it may not actually be any easier to get into the good institutions if you already have a PhD. Finally, even if you do, you will be 5 years older than everyone else when you graduate, and this will count against you (statistically speaking) when you look for jobs beyond that.

The thing you have come to realize is that everyone wants to go to MIT and Stanford, but only few actually manage to. Make your peace with this -- shoot for the best place you can to get a PhD and then make your career from there. You'll find that with a PhD from a good place in Europe, you will have sufficiently many options.