The current record is ten trillion and fifty digits. ^[1]

The last known digit is 1 .

How many digits of $\pi$ are currently known? Well, all of them!

It is possible to compute the $n^{\text{th}}$ digit of $\pi$ without computing the preceeding $n-1$ digits first! See, for example, Wikipedia ^[1].

The problem with the question as stated, I believe, comes down to defining "known". @Xnyyrznaa makes an excellent point in the comments above when he says "nobody remembers the ten trillion...they are not written down on paper." So, in some ways, skullpatrol's answer is somewhat unsavoury. We "know" that the ten-trillion-and-fiftieth digit is a one, but we do not "know" the rest; not by any reasonable metric (that is, according to his answer).

The concept of "know" corresponds to a function. In life, because everything is finite, this function is simple - it is the "look it up on a list" function (and is the function skullpatrol is getting at). In mathematics, this function can be pretty exotic. My point is that a much better function exists for finding a specific digit of $\pi$.

For instance, Wikipedia ^[2] tells us that the five-trillionth, 40-trillionth and quadrillionth digits of $\pi$ are all zero.

Also I think that this is sort of misconception that how many digits oh $\pi$ are known. $\pi$ is a number in itself, whose digits need not be known as it is an irrational number, a type of undefined abstraction which is used to model certain things in our nature, like the length of hypotenuse of a unit isoceles right triangle. But what can be known is a rational number of the form $x/10^n$ where x,n belong to the set of integers that is nearer to $\pi$ compared to all other rational numbers of that form.