Suppose that the Federal Reserve did not conduct any open market operations and allowed interest rates to freely float, while the treasury continued borrowing to fund the budget deficit. What would interest rate (on treasury bonds let's say to be specific) be? I am trying to understand how artificial current interest rates currently are.
While Nathan is correct above that government actors, not just the Fed but all actors including foreign, will likely never stop monitoring and manipulating the US interest rates. This doesn't mean that the market is not deciding the price! While the Fed is clearly has a lot of influence in this market they are but one participant and while they do change the yields the value does currently freely float. It is not set to a value by law.
This means your question can be answered as long as you are willing to take into account that the market always includes the possibility of future intervention by government actors domestic and foreign.
Long-term interest rates (10-30 years) include a lot of time where the current intervention will have likely stopped. Rates that far out are currently 2.4-3.0%. However, this is not quite the right answer either as the 10 year value, for instance, includes at least a year more of open market operations and some time effects for locking your money up for that long.
To correct for this you can calculate forward interest rates which are essentially the expected short term rates in the future beyond the period. This can give you a feel for what the market expects the interest rates to become after the current intervention. For instance, the 1 year rate today is around 0.11% while the one year rate a year from now is expected to be 0.40% or so.
Choose a time after you think the current interventions will be completed (three years?) and it is really easy to calculate.