Well in computing it can go either way, and it often will, even within the same system.
Intel CPUs as an example are "little endian" meaning that the bit with the lowest address is the least significant digit. So, in your example, this would be 1-2-4-8.
ARM CPUs were big-endian (8-4-2-1) but as of version 3 they are now bi-endian, meaning that you can control the endianness via software configuration to suit your own needs. Very handy.
When you design hardware that is implemented on an FPGA, it is very easy to swap endianness or to use little endian everywhere when you wanted big endian.
In short, both types are used all over the freaking place in digital systems.