This question arose when I heard about Mill's constant: the number $A$ such that $\lfloor A^{3^n} \rfloor$ is prime for all $n$. It made me wonder whether $A^{3^n}$ could be made arbitrarily close to composite numbers, or in other words: "how near can $A^{3^n}$ get to the 'next integer'?"
I had a feeling, perhaps erroneous, that this might have to do with the rationality of $A$, but since this is an open question, I decided to consider the more general question in the title.
Having very little knowledge in this area, the only two thoughts that occurred to me were:
to write $q=k+\epsilon$, where $\epsilon<1$ and $k\in\mathbb{N}$ and consider how $k$ and $\epsilon$ effect each other in $(k+\epsilon)^n$. The appears to be difficult because of the symmetry in the binomial expansion. This is also where the rationality of $\epsilon$ might come into play.
to see whether $q^{-n}$ might get arbitrarly close to the terms of the harmonic sequence since they both converge to the same thing, $0$. This one is probably a bit naive.
A particular class of algebraic integers is called Pisot numbers. These numbers have the property you describing.
A Pisot number is an algebraic integer greater than $1$ which has all of its conjugates inside the unit circle. Or equivalently if $f$ is an irreducible polynomial with integer coefficients and all of its roots except one are inside the unit circle, then the root outside the unit circle is a Pisot number. A Pisot number is never an integer.
Let $\theta$ be a pisot number and $c_1,\ldots,c_n$ its conjugates. From the theory of the symmetric polynomials it follows that $$ \theta^k+\sum_{i=1}^nc_i^k\in\mathbb Z, \; \forall k\in\mathbb N. $$ Since $|c_i|<1$ it follows that $c_i^k\xrightarrow[k\to\infty]{}0 $. Therefore for large $k$, $\theta^k$ is very close to an integer.
The smallest Pisot number is $\theta_0=1.3247179572447460260$.
In the following table I list the values of $\theta_0^k$ for $k=1,2,\ldots,50$. $$ \begin{array}{|c |c |c |c |} \hline k & \theta_0^k & k & \theta_0^k\\ \hline 1 & 1.324717957 & 26 & 1496.955904 \\ \hline 2 & 1.754877666 & 27 & 1983.044367 \\ \hline 3 & 2.324717956 & 28 & 2626.974482 \\ \hline 4 & 3.079595621 & 29 & 3480.000269 \\ \hline 5 & 4.079595620 & 30 & 4610.018846 \\ \hline 6 & 5.404313575 & 31 & 6106.974748 \\ \hline 7 & 7.159191238 & 32 & 8090.019112 \\ \hline 8 & 9.483909190 & 33 & 10716.99359 \\ \hline 9 & 12.56350481 & 34 & 14196.99385 \\ \hline 10 & 16.64310042 & 35 & 18807.01269 \\ \hline 11 & 22.04741399 & 36 & 24913.98743 \\ \hline 12 & 29.20660521 & 37 & 33004.00653 \\ \hline 13 & 38.69051439 & 38 & 43721.00011 \\ \hline 14 & 51.25401918 & 39 & 57917.99394 \\ \hline 15 & 67.89711957 & 40 & 76725.00660 \\ \hline 16 & 89.94453353 & 41 & 101638.9940 \\ \hline 17 & 119.1511387 & 42 & 134643.0005 \\ \hline 18 & 157.8416530 & 43 & 178364.0005 \\ \hline 19 & 209.0956721 & 44 & 236281.9944 \\ \hline 20 & 276.9927916 & 45 & 313007.0009 \\ \hline 21 & 366.9373250 & 46 & 414645.9947 \\ \hline 22 & 486.0884635 & 47 & 549288.9950 \\ \hline 23 & 643.9301163 & 48 & 727652.9952 \\ \hline 24 & 853.0257881 & 49 & 963934.9892 \\ \hline 25 & 1130.018579 & 50 & 1276941.990 \\ \hline \end{array} $$