[**Edited** mostly to extend the computation from $1.5 \cdot 10^{13}$
to a bit over $2^{50} > 10^{15}$ and give the heuristics for expected number of
records for $\| r(n) \|$ vs. $\log n \cdot \| r(n) \|$]

Just ran across this. I see that Kevin's answer completely settles the
original question, but meanwhile Will Jagy raised the question of
finding new record lows for
$$
\log n \cdot \left\| \frac{n}{\log n} \right\|
$$
and proving their infinitude. I next outline a proof that there are
infinitely many such record lows, and then report on a computation of
all such $n$ up to $1.5 \cdot 10^{13}$.

For the infinitude: Since $r(n) := n / \log n$ can never be an exact integer,
it is enough to prove that for each $\epsilon > 0$
there exist infinitely many solutions of $\| r(n) \| < \epsilon/\log n$.
In fact it's not hard to show that $\| r(n) \|$ can get as small as some
negative power of $n$, because $r(n)$ is almost linear
(its second derivative is $o(n^{-1})$ as $n \rightarrow \infty$)
and we can choose $n_0$ to make $r'(n_0)$ as far as possible from
any rational number. If I did this right, we can find intervals
$|n - n_0| \leq h$ in which $\min_n \| r(n) \| \ll h^{-1}$
where $h^{-1} = |r''(n_0)|^{1/3} \sim (n_0 \log^2 n_0)^{-1/3}$.
For instance, we may choose $n_0$ so that
$r'(n_0) = 1 / (k + \sqrt 2)$ for $k = 1, 2, 3, \ldots$
[that is, so that $\log n_0$ solves the quadratic equation
$\lambda^2 = (k+\sqrt2) (\lambda-1)$].
On such an interval, $r(n)$ is approximated by $r(n_0) + r'(n_0)(n-n_0)$
to within $O(r''(n_0) (n-n_0)^2) = O(h^2/h^3) = O(h^{-1})$, and
(since $h$ grows much faster than $k$)
the arithmetic sequence with common difference $r'(n_0)$
is close enough to being equidistributed that it comes within
$O(1/h)$ of an integer. [We probably expect that $\| r(n) \|$
is random enough that it gets as small as $c/n$ or even $o(1/n)$,
but proving such a result must be well out of reach.]

For the numerical search, the problem is quite similar to
MO.19170
on nearly-integral values of $\log_{10} n!$ (since $n/ \log n$,
like $\log_{10} n!$, is nearly linear in $n$). Again it takes time
only $\tilde O(N^{2/3})$ to find all examples with $n < N$ using
a linear-approximation technique such as described at the bottom of
page 15 of
Lefèvre's slides.
This is actually the same idea as in the previous paragraph:
partition $[1,N]$ into intervals $|n-n_0| \leq h \sim (n_0 \log^2 n_0)^{1/3}$;
in general $r'(n_0)$ might be so close to a rational number that
equidistribution fails, but we can still use continued fractions
to find all $n$ in that interval for which $\|r(n)\| \ll h^{-1}$.

I ran this with $N = 2^{50} > 10^{15}$ on ten **alhambra** heads.
Most finished in under two days; two took an extra day or two,
probably spending most of them on $n_0$ for which
$r'(n_0)$ was nearly rational (in this case one can do much better than
trying every $n \in [n_0-h,n_0+h]$ for which
$\| r(n_0) + r'(n_0)(n-n_0) \|$ is small,
but I didn't take the extra time to implement that refinement).
The computation found fourteen new records beyond the
12 initial terms 2, 17, 163, 715533, 1432276, 6517719, 11523158,
11985596, 24102781, 254977309, 451207448, 1219588338
of OEIS sequence A178806, namely

2048539023, 10066616717, 42116139191, 47657002570,
73831354169, 122478947521, 143949453227, 3152420311977,
5624690531099, 14964977749017, 25999244327633, 92799025313425,
164330745650026, and 604329910739082.

There is also a new example, namely $n = 3040705645816$, of a number
that is not in this sequence but does belong in the closely
related OEIS sequence A178805,
which consists of $n$ that achieve record low values of $\| r(n) \|$
instead of $\log n \cdot \| r(n) \|$. In general a
$\log n \cdot \| r(n) \|$ record is automatically also an
$\| r(n) \|$ record, but the converse can fail on occasion.
If we imagine that the $\| r(n) \|$ are independent random numbers
uniformly distributed on $(0,1/2)$ then the probability that
$\| r(n) \|$ is a new record is $1/n$, so we expect $\log N + O(1)$
record values with $n \leq N$. The same question for
$\log n \cdot \| r(n) \|$ is trickier, but if I did this right
the probability that $\| r(n) \|$ is a new record but
$\log n \cdot \| r(n) \|$ is *not* one is approximately $1 / n \log n$,
so we expect only $\log\log N + O(1)$ examples such as
$n = 3040705645816$ up to $N$, and might never see another one
even though there should be infinitely many more.

Here is a table of the values of $n < 2^{50}$
for which $\| r(n) \|$ attains a new record low, together with
the signed fractional part of $r(n)$, and $\log n$ times that fractional part:
$$
\begin{array}{rrrc}
2 & -0.1146099 & -0.0794415 & \\
5 & 0.1066747 & 0.1716863 & ! \\
9 & 0.0960765 & 0.2111017 & ! \\
13 & 0.0683262 & 0.1752532 & ! \\
17 & 0.0002541 & 0.0007199 & \\
163 & -1.26 \cdot 10^{-6} & -6.43 \cdot 10^{-6} & \\
53453 & 1.22 \cdot 10^{-6} & 1.33 \cdot 10^{-5} & ! \\
110673 & 6.68 \cdot 10^{-7} & 7.76 \cdot 10^{-6} & ! \\
715533 & 3.84 \cdot 10^{-7} & 5.17 \cdot 10^{-6} & \\
1432276 & 2.33 \cdot 10^{-7} & 3.30 \cdot 10^{-6} & \\
6517719 & -2.00 \cdot 10^{-7} & -3.14 \cdot 10^{-6} & \\
11523158 & -9.95 \cdot 10^{-8} & -1.62 \cdot 10^{-6} & \\
11985596 & -7.26 \cdot 10^{-8} & -1.18 \cdot 10^{-6} & \\
24102781 & 4.43 \cdot 10^{-9} & 7.53 \cdot 10^{-8} & \\
254977309 & 9.12 \cdot 10^{-10} & 1.76 \cdot 10^{-8} & \\
451207448 & 3.68 \cdot 10^{-10} & 7.33 \cdot 10^{-9} & \\
1219588338 & -2.57 \cdot 10^{-10} & -5.38 \cdot 10^{-9} & \\
2048539023 & -5.89 \cdot 10^{-11} & -1.26 \cdot 10^{-9} & \\
10066616717 & 4.85 \cdot 10^{-11} & 1.12 \cdot 10^{-9} & \\
42116139191 & -4.47 \cdot 10^{-11} & -1.09 \cdot 10^{-9} & \\
47657002570 & -2.43 \cdot 10^{-11} & -5.97 \cdot 10^{-10} & \\
73831354169 & 1.35 \cdot 10^{-11} & 3.38 \cdot 10^{-10} & \\
122478947521 & 7.53 \cdot 10^{-13} & 1.92 \cdot 10^{-11} & \\
143949453227 & -5.50 \cdot 10^{-13} & -1.41 \cdot 10^{-11} & \\
3040705645816 & 5.18 \cdot 10^{-13} & 1.49 \cdot 10^{-11} & ! \\
3152420311977 & -3.36 \cdot 10^{-13} & -9.67 \cdot 10^{-12} & \\
5624690531099 & 1.28 \cdot 10^{-13} & 3.76 \cdot 10^{-12} & \\
14964977749017 & -7.15 \cdot 10^{-14} & -2.17 \cdot 10^{-12} & \\
25999244327633 & -2.02 \cdot 10^{-14} & -6.25 \cdot 10^{-13} & \\
92799025313425 & 6.01 \cdot 10^{-15} & 1.93 \cdot 10^{-13} & \\
164330745650026 & -1.00 \cdot 10^{-15} & -3.28 \cdot 10^{-14} & \\
604329910739082 & -4.59 \cdot 10^{-16} & -2.27 \cdot 10^{-14} &
\end{array}
$$
the "!"'s mark the $\| r(n) \|$ records that aren't $\log n \cdot \| r(n) \|$ records.

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