Let $\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly $59.99^\circ$, $60^\circ$, and $60.01^\circ$. For each positive integer $n$, define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-1}C_{n-1}$, and $C_n$ to be the foot of the altitude from $C_{n-1}$ to line $A_{n-1}B_{n-1}$. What is the least positive integer $n$ for which $\triangle A_nB_nC_n$ is obtuse?

Here the initial perturbation is $a_0=0.01$ (from $59.99,60,60.01$). As before, the perturbation doubles each iteration, so we need the least $n$ with $60+2^n(0.01)>90$, i.e. $2^n>3000$.

$2^{11}=2048<3000<4096=2^{12}$, so the least such $n$ is $\boxed{12}$.