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##Pyth, 15 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 (`) is conveniently [].

Try it here.

Pyth, 15 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 (`) is conveniently [].

Try it here.

##Pyth, 15 13 bytes

-2 bytes by @Maltysen

eSm-F/Ld`Y._z

Counts the difference of [s and ]s, and takes the maximum.

Try it here.

##Pyth, 15 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 (`) is conveniently [].

Try it here.

##Pyth, 15 bytes

eSm-F/Ld"[]"._z

Counts the difference of [s and ]s, and takes the maximum.

Try it here.

##Pyth, 15 13 bytes

-2 bytes by @Maltysen

eSm-F/Ld`Y._z

Counts the difference of [s and ]s, and takes the maximum.

Try it here.