If you have an integer of $n$ decimal digits that is odd and non-divisble by $5$ then what is the shortest repeating decimal it can have in its reciprocal? Can it be as small as $n^c$ digits for some fixed $c>0$?
In other words what is the smallest factor of the Carmichael function of an integer of magnitude $>10^n$ that can occur as a period if the integer is coprime to $10$?
I am looking for an analytic statement such as 'if you pick a random decimal of $n$ digits coprime to $10$ then with probability $p$ it has repeating inverse of length at most $n^c$'.
Theorem 1 here seems to say there are integers of $n$ decimal digits with carmichael function as small as $n^{c_2\log\log n}$ (http://www.math.drexel.edu/~eschmutz/PAPERS/lambda.pdf). Can we we find at least one coprime pair of $n$ decimal digits with carmichael function this small?
I am mostly looking for how small we can get and not how large.
In the worst case (as pointed out in the comments), the smallest period of the reciprocal of a number with $n$ decimal digits is $n$ (achieved for $\frac{1}{10^{n}-1}$). No number with $n$ decimal digits can have a smaller period, since the first nonzero entry in its reciprocal must occur in the $n$th position.
The question for average case is much harder, and depends on the resolution to Artin's conjecture. Artin's conjecture states that, if $a$ is not a square or $-1$, then $a$ is a primitive root modulo roughly 37% of the primes. Applying this to $a=10$, this means if you take a random $n$-digit prime number $p$, the period of its reciprocal is $p-1$ (which is on the order of $10^n$) with probability at least 37%. By the prime number theorem, the probability a random $n$-digit number is prime is at least $\Omega(1/n)$. Together, these facts imply that the average period length over all $n$-digit numbers is at least $\Omega(10^{n}/n)$, much larger than the polynomial bounds you propose above (probably with some extra work you can improve this to $\Omega(10^{n})$).
(Of course, this is all conditioned on Artin's conjecture. It's possible there's some non-conditional approach to show the same bounds, but I haven't been able to think of one yet.)