Calculating a SQRT digit-by-digit? I need to calculate the SQRT of  $x$ to $y$ decimal places. I'm dealing with $128$-bit precision, giving a limit of $28$ decimal places. Obviously, if $\,y > 28$, the Babylonian method, which I'm currently using, becomes futile, as it simply doesn't offer $y$ decimal places.
My question to you is, can I calculate the Square Root digit-by-digit? For example, calculating the $n$-th digit, assuming I already have digit $\,n - 1$. I'm well aware this may be impossible, as I've not found any such method through Google yet, just checking here before I give up in search of arbitrary precision methods.

As explained by two other answers, there is a method that yields the decimal places one by one. Unfortunately for you, the method still requires you to store and manipulate an approximation and the associated increment. Your problem is that you can only store them to 28 decimal places, so you will still not be able to go very far.
You can't have "arbitrary precision"; you will need to decide how many decimal places you are going to have. Suppose you decide on 100 dp. Since 100/28 is a little less than 4, you will need at least four 128 bit numbers to represent a single number. You will also need to create routines for operating on such numbers that are combinations of other numbers. As one of the operations you need to perform is multiplying, you may have difficulties when multiplying two 28 dp numbers. For that reason you could simplify some stages by using your (potentially) 28 dp numbers to hold number sthat never exceed 14 dp, except temporarily in the middle of a calculation. 
