##Pyth, <s>15</s> 13 bytes

*-2 bytes by @Maltysen*

    eSm-F/Ld`Y._z

Counts the difference between the cumulative counts of `[` and `]`, and takes the maximum. `Y` is the empty array, and its string representation (<code>`</code>) is conveniently <code>[]</code>.

Try it [here](http://pyth.herokuapp.com/?code=eSm-F%2FLd%60Y._z&input=%5B1%2C+%5B%5B2%2C+3%2C+%5B%5B4%5D%2C+5%5D%2C+6%2C+%5B7%2C+8%5D%5D%2C+9%2C+%5B10%2C+%5B%5B%5B11%5D%5D%5D%5D%2C+12%2C+13%5D%2C+14%5D&debug=0).