  Extreme mutation rates in microbes and cancer cells can result in
error-induced extinction (EEX), where every descendant cell eventually acquires
a lethal mutation. In this work, we investigate critical birth-death processes
with $n$ distinct types as a birth-death model of EEX in a growing population.
Each type-$i$ cell divides independently $(i)\to(i)+(i)$ or mutates
$(i)\to(i+1)$ at the same rate. The total number of cells grows exponentially
as a Yule process until a cell of type-$n$ appears, which cell type can only
die at rate one. This makes the whole process critical and hence after the
exponentially growing phase eventually all cells die with probability one. We
present large-time asymptotic results for the general $n$-type critical
birth-death process. We find that the mass function of the number of cells of
type-$k$ has algebraic and stationary tail $(\text{size})^{-1-\chi_k}$, with
$\chi_k=2^{1-k}$, for $k=2,\dots,n$, in sharp contrast to the exponential tail
of the first type. The same exponents describe the tail of the asymptotic
survival probability $(\text{time})^{-\chi_n}$. We present applications of the
results for studying extinction due to intolerable mutation rates in biological
populations.
