Eh, it's kind of like the decision on how bags of groceries to carry from the car to the house at once -- it's faster the more you take, but the more you carry the more likely you are to drop them and the more catastrophic the accident becomes when it happens.
It's tricky to take large bites here because of the short doubling time compared to the relatively longer recovery time -- in the ~3 weeks it takes your currently-infected cohort to recover, your infected population could increase 100-fold if you aren't careful. Mistiming things -- misjudging how many people are currently infected -- by a couple of days can be difference between exposing 10% of your population to the risk and exposing 15%. There's a reason pandemics are often compared to forest fires.
This will get easier as time goes on, as the doubling period gets longer and longer.
Based on current estimates of how many people will die in the US, you can infer that we currently expect 5-10% of the population to be exposed in this first wave; I believe what widespread testing there has been in Italy leans towards about 10% of that population having been exposed there as well.