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.
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