I searched numbers $N$, such that the continued fraction of $N^{1/3}$ has very large entries. I only searched for a single large entry, but I was surprised that two continued fractions contained not only one large entry, but multiple large entries. Here the two amazing expansions :
$$102175 [46, 1, 2, 1, 8741, 2, 186, 2, 13112, 1, 6, 1, 8, 2, 9, 2, 623, 1, 33, 1 , 9, 1, 2, 2, 17484, 14, 2, 2, 1, 4, 19021, 2, 1, 1, 1, 1, 1, 1, 3437888, 2, 2, 6, 21510, 2, 1, 2, 55063048, 1, 1, 1, 1, 1, 2, 8, 44, 2, 1, 4, 1, 4, 61, 2, 1666 1, 2, 1, 3, 1, 1, 23, 1, 4, 2, 2, 8, 3, 3, 1, 1, 2, 6, 3, 1, 1, 3, 5, 17, 21, 17 , 3, 168, 3, 1, 1, 17, 1, 3, 2, 3, 4, 3] 55063048$$ $$267090 [64, 2, 2, 31104, 1, 4, 64, 4, 1, 46657, 1288, 55545, 1127, 62210, 2, 2, 40, 1, 1, 2, 1, 1, 4, 559, 8, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 1, 101091, 3, 1, 1, 8, 6, 10, 3, 1, 2, 2, 1, 1, 2, 17, 2, 1, 1, 2, 1, 4897902700, 1, 54, 1, 288, 1, 1, 1, 20, 1, 1, 5, 31360929, 1, 15, 9, 1, 1, 30, 1, 5, 6, 2, 7, 16, 2, 3, 1, 2, 3, 9935, 1, 3, 2, 1, 5, 4, 4, 2, 1, 28, 1, 27] 4897902700$$
Explanation : First, the number $N$ is displayed, then the first $100$ terms of the continued fraction of $N^{1/3}$ and finally the maximum of the entries.
The cubic roots of the numbers $102175$ and $267090$ seem to have a very special continued fraction.
In the second continued fraction, we have even $5$ consecutive large entries, and in both continued fractions we have an entry larger than $10^6$ besides the maximum entry.
This is not at all what I expected, in particular because almost every real number has a continued fraction expansion that follows a special distribution mentioned here :
Typicality of boundedness of entries of continued fraction representations
The continued fraction expansions above are far away from this distribution (even if we do not consider the maximum entry).
How can this phenomen be explained ?
This does not answer the question, but it may give some ideas as to where to look for an answer.
In 1965, Brillhart found that the real root of $x^3-8x-10$ has several very large partial quotients in its continued fraction expansion, e.g.,
$a_{17}=22986$
$a_{33}=1501790$
$a_{59}=35657$
$a_{81}=49405$
$a_{103}=53460$
$a_{121}=16467250$
$a_{139}=48120$
$a_{161}=325927$
After this, it "calms down". There is an explanation for this behavior, involving some fairly advanced stuff (modular forms and the like). See, for example,
Churchhouse and Muir, Continued fractions, algebraic numbers and modular invariants, J. Inst. Maths Applics 5 (1969) 318-328.
Stark, An explanation of some exotic continued fractions found by Brillhart, in Atkin and Birch, eds., Computers in Number Theory, Academic Press 1971, 21-35.
https://oeis.org/A002937 is also worth a look, and possibly R.P. Brent, Alfred J. van der Poorten, Herman J.J. te Riele: "A comparative study of algorithms for computing continued fractions of algebraic numbers" Algorithmic Number Theory (Talence, 1996), pages 35-47, Lecture Notes in Computer Science, 1122, Springer, Berlin, 1996.