Add a new element to an array without specifying the index in Bash

Yes there is:

ARRAY=()
ARRAY+=('foo')
ARRAY+=('bar')

Bash Reference Manual:

In the context where an assignment statement is assigning a value to a shell variable or array index (see Arrays), the ‘+=’ operator can be used to append to or add to the variable's previous value.

As Dumb Guy points out, it's important to note whether the array starts at zero and is sequential. Since you can make assignments to and unset non-contiguous indices ${#array[@]} is not always the next item at the end of the array.

$ array=(a b c d e f g h)
$ array[42]="i"
$ unset array[2]
$ unset array[3]
$ declare -p array     # dump the array so we can see what it contains
declare -a array='([0]="a" [1]="b" [4]="e" [5]="f" [6]="g" [7]="h" [42]="i")'
$ echo ${#array[@]}
7
$ echo ${array[${#array[@]}]}
h

Here's how to get the last index:

$ end=(${!array[@]})   # put all the indices in an array
$ end=${end[@]: -1}    # get the last one
$ echo $end
42

That illustrates how to get the last element of an array. You'll often see this:

$ echo ${array[${#array[@]} - 1]}
g

As you can see, because we're dealing with a sparse array, this isn't the last element. This works on both sparse and contiguous arrays, though:

$ echo ${array[@]: -1}
i

$ declare -a arr
$ arr=("a")
$ arr=("${arr[@]}" "new")
$ echo ${arr[@]}
a new
$ arr=("${arr[@]}" "newest")
$ echo ${arr[@]}
a new newest