We begin by listing the smallest positive perfect cubes with two digits or fewer:

1, 8, 27, 64.

And now we sum them.  $1+1+1$ is too small; $1+1+8=10$ which is not prime; but $1+8+8=17$, which is indeed a prime.  Hence the answer is $\boxed{17}$.
What is the smallest two-digit prime that is the sum of three (not necessarily distinct) positive perfect cubes?
{'answer': '17', 'subject': 'Prealgebra', 'level': 5, 'unique_id': 'test/prealgebra/1982.json'}