Is there an example where the likelihood principle would lead to markedly different (and equally defensible) inferences?

All the examples I see are comparing a binomial with a negative binomial, where the p-value of the first is 7% and of the second 3%, which are markedly different only if one is making binary decisions on arbitrary thresholds of 5%.

I've never seen an example where it would lead to marked different and both defensible inferences.