Let's consider this dense packing scheme, which is shown in the attached diagram.

First horizontal row of bottles will have  990 : 9 = 110 bottles.

The line of their centers is y = 4.5 cm.


The centers of the circles form a grid of equilateral triangles with the side length of 9 cm.


When the side of an equilateral triangle is a = 9 cm, its height/altitude is  

    h =  =  = 7.79423 cm (rounded up).


So, the second row of circles is 7.79423 cm above the first horizontal line y = 4.5 cm.

Thus, the equation of the 2nd horizontal line of centers is  y = 4.5 + 7.79423,  or  y = 12.29423 cm.


The most upper horizontal line of circles is not closer than 4.5 cm to the upper bound y = 990 cm.


Therefore, if the total number of horizontal lines of centers is n, we can write this inequality

    4.5 + (n-1)*7.79423 + 4.5 <= 990  cm,

or

    (n-1)*7.79423 <= 990 - 4.5 - 4.5 = 990 - 9 = 981 cm.


S0, the number of lines is about (is not more than)

    n <=  + 1 = 125.8623366 + 1.


Rounding to the closest lesser integer number, we see that

    n <= 126.


We see that there is the room for 126 lines of centers, but there is no room for 127 lines.


The number of bottles / (circles) in lines is altering  110, 109, 110, 109, . . . , 110, 109, 110, 109.


So, this sequence has 63 lines with 110 bottles and 63 lines with 109 bottles.


Hence, the total maximum possible number of bottles is  63*110 + 63*109 = 13797.


ANSWER.   The total maximum possible number of bottles in the box is  13,797.