Greedy Pierce expansions of an irrational number, with the constraint that the terms must always increase, tend to have terms that grow exponentially. In contrast, the continued fraction expansion does not grow like that.
However, I suspect that by dropping the increasing constraint, and not being greedy, there exist Pierce (or Pierce-like) expansions which remain small.
Like continued fractions, terms alternate between over and under estimating. Each term multiplies in to create ever smaller denominators.
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