= Solution
For independent equiprobable bases there are $4^3=64$ <Codon>[codons], three of them stops, so $p_{\mathrm{stop}}=3/64$. In a fixed <reading frame>, and conditional on starting <translation (biology)>[translation], surviving at least $L$ successive non-stop <Codon>[codons] has <probability> $\boxed{(61/64)^L}$. For 300 it is about $5.6\times10^{-7}$. Strictly “exceeding 300” requires 301 non-stop <Codon>[codons], giving $(61/64)^{301}$. A specified <start codon>, <codon> bias, correlations or looking for a long sequence anywhere in a large genome changes the sampling problem and cannot be silently folded into this single-frame <probability>.
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