Consider this example: I have 3 divisors $D_1, D_2, D_3$ in a variety $X$, let's say for simplicity that they have simple normal crossing. Now let's say I blow up the intersection $Z_1 = D_2 \cap D_3$, and denote my new variety by $X'$ with blow up morphism $\pi: X' \rightarrow X$. In the next step, I could do the "same" thing and blow up the intersection of the strict transforms $Z'_3 = D'_1 \cap D'_2$, denote this new space by $X''$. WHat if I want to get from $X$ to $X''$ "directly? Is there a blowup which would describe $X'' \rightarrow X$ ? Vice versa, if i have a blow up with multiple exceptional divisors, can I replace that by a sequence of blowups?
I have been sketching the dual complexes of these divisors (imagine 3 coordinate planes) and from my sketches, I cannot see the relationship between the simultaneos and sequence of blowups, but maybe I am wrong?