First, I'd like to revisit Will Sawin's upper bound, but I make no promises about off-by-one errors (which I think I made in comments elsewhere on this page).  The product of any sequence of $x$ consecutive integers has $p$-valuation at least $\lfloor \frac{x}{p-1}\rfloor - \lfloor \log_p(x+1) \rfloor$.  The $p$-valuation of $\binom{n}{a}$ is the number of carries when adding $a$ to $n-a$ in base $p$, so this is bounded above by $\lfloor \log_p n \rfloor$.  In other words, the product of the $x = b-(n-a)$ integers from $n-a+1$ to $b$ must satisfy $\lfloor \frac{x}{p-1}\rfloor - \lfloor \log_p(x+1) \rfloor \leq \lfloor \log_p n \rfloor$.  Pushing some terms around yields the bound $$x \leq (p-1)\lfloor log_p(n) \rfloor +(p-1) \lfloor log_p log_p n \rfloor + 2p - 2.$$
If we only worry about making $\frac{n!}{a!b!}$ into a 2-adic integer instead of an ordinary integer, then this bound is sharp, since the case $n=2^k, a=2^k-1, b = 2^y-1$ saturates it when $k = 2^y-y-1$.

When $n$ is large, $(p-1) \lfloor \log_p(n) \rfloor > \lfloor \log_2(n) \rfloor$ for odd $p$, so this suggests that it is good to check 2-valuations when eliminating candidate triples.  By request, here is some computer code.

**Edit:** Following Noam D. Elkies's suggestion, I rewrote my program in C.  Also, I found an overflow bug in my valuation routine, as a result of using 32 bit integers, so some of the previous results for $n>65536$ were false.  Since the C code is somewhat longer than the SAGE version, I put it [here on my web page][1].

Anyway, starting with k=3 and running up to $2^{23} = 8388608$ took about 3 hours on a 12-core desktop.  Here are the results in the form $a, b, n, k$:

    3, 5, 6, 2
    6, 7, 10, 3
    11, 29, 36, 4
    14, 47, 56, 5
    47, 59, 100, 6
    59, 110, 162, 7
    23, 241, 256, 8
    31, 239, 261, 9
    187, 239, 416, 10
    447, 620, 1056, 11
    239, 1853, 2080, 12
    1439, 1775, 3201, 13
    1663, 2735, 4384, 14
    14335, 18458, 32778, 15
    9209, 23615, 32808, 16
    13106, 52447, 65536, 17
    26207, 39347, 65536, 18
    34719, 227444, 262144, 19
    109237, 152927, 262144, 20
    59039, 465398, 524416, 21
    230111, 294839, 524928, 22
    496123, 3698204, 4194304, 23
    1007871, 3186457, 4194304, 24
    983546, 7405087, 8388608, 25
    1029947, 7358687, 8388608, 26

I checked a few larger powers of two - here are the maximal values of $a+b-n$:

    2, 3, 4, 1
    2, 7, 8, 1
    5, 14, 16, 3
    9, 26, 32, 3
    19, 49, 64, 4
    17, 116, 128, 5
    23, 241, 256, 8
    53, 467, 512, 8
    314, 719, 1024, 9
    482, 1574, 2048, 8
    1943, 2164, 4096, 11
    1979, 6227, 8192, 14
    3317, 13081, 16384, 14
    6548, 26234, 32768, 14
    26207, 39347, 65536, 18
    56375, 74354, 131072, 17
    109237, 152927, 262144, 20
    58559, 465749, 524288, 20
    117350, 931247, 1048576, 21
    511613, 1585561, 2097152, 22
    1007871, 3186457, 4194304, 24
    1029947, 7358687, 8388608, 26
    839773, 15937469, 16777216, 26
    2479487, 31074973, 33554432, 28
    28499711, 38609183, 67108864, 30
    13640318, 120577439, 134217728, 29
    26335607, 242099879, 268435456, 30
    149450463, 387420479, 536870912, 30
    42571871, 1031169982, 1073741824, 30

  [1]: http://www.math.tsukuba.ac.jp/~carnahan/MO2.c