I was struck by a comment someone made about the recent BB(748) ZFC-independence result being shortened to BB(745) pointing out that while this reduced the number of TM states by only 3 it reduced the value of BB(n) by a really big factor.
Given that we have a few values and estimates for discrete BB(n) values (Defining BB to be the shift-count function here but I'm not sure that matters)...
$BB(1)=1, BB(2)=6, BB(3)=21, BB(4)=107, BB(5)\ge47,176,870\;\;\text{and}\;\;BB(6)>10\uparrow\uparrow15$
Can we say (with any confidence) the following?
$\lim\limits_{n\rightarrow\infty}\frac{BB(n)}{BB(n-1)}=\infty$
...and...
$\forall{n\ge3}:\;\;\frac{BB(n+1)}{BB(n)}>\frac{BB(n)}{BB(n-1)}$
Or might there be moderately (for BB) relatively flat plateaus interspersed between effectively vertical cliffs? BB(2) = 6BB(1) but BB(3) is only $\frac{21}{6}$BB(2); after this normal service seems to be resumed.