What should be the market cap of bitcoin in your opinion if we want the bitcoin price to be relatively stable and saved from manipulation ?
Normally the stability of a currency mainly come from the monetary policies created, followed and managed by the owners of it. Which are the States of all the world. They delegate such power to one monetary Authority which normally is called Central Bank (but not always so; in USA for example is called Federal Reserve). The lasts take care and manage only the national currency. All their work and the structure they have is build to serve this aim. Normally, theoretically, the amount of money in circulation in a given moment must be equals with the overall value of all the good produced and in market and part of those in production in that given moment plus an amount which allow a desired amount of inflation of that currency (all this is very simplified).
Bitcoin have nothing similar. It is a pure product of market which move and play according to the rules and the "caprices" of the free market. Have no owners which can take care of it, have no goods which can justify its existence on the market. Have one particularity which make it very strong. Its flow in the market is constant. So in this case we have an overturned situation compared to the above explained. In the above the money must "follow" the value of goods in the market while in the case of bitcoin is the flow of the amount of bitcoin in the market (which is fixed) which must be followed by everything. This flow, being fixed, is the initial point of everything has to do with bitcoin. With this situation is hard to tell that bitcoin can follow the normal rules followed by the normal currencies in use and issued by the Central Banks.
So no one can know when or if someday bitcoin will have a stable price or less volatile possible. Can be guessed that this may happen with its maximal spread and the possible increase of price. But are only guesses. I, for example, personally believe that bitcoin will have one day a very high price and will be an almost stable currency when it will be in these levels. This is one my hope and maybe conviction given from the facts that it is the first materialized product of a disruptive technology (peer to peer), and above all, because it is in limited number. But i am sure that this cannot be verified if bitcoin will be not regulated by most or majority of the countries. The non legalization will transmute it in a marginal currency mainly used for illegal stuffs and only from people devoted to it. That's all I can tell about your question.