Repeat every other character in string starting with second character

sed, 12

s/.(.)/&\1/g

Try it online!

brainfuck, 13 or 11 bytes

>,[.,[..,>]<]

Assumes EOF->0. :(

You can try it online!

(Assuming input also continues to give zeroes indefinitely after the first EOF, we can do it in 11 bytes:

,[.,[..>],]

though this eats memory.)

Jelly, 3 bytes

ḤÐe

A full program accepting a string which prints the result

Try it online!

How?

Utilises the fact that the implementation of the double atom, Ḥ, is multiplication by two, and that strings are lists of Python characters. In Python when a character is multiplied by two it becomes a Python string of length two (e.g. 'x'*2=='xx'). Lastly the implicit output of a list composed entirely of characters and strings (or lists thereof) is its content all smashed together.

ḤÐe - Main Link: list of characters  e.g. input abcd -> ['a','b','c','d']
 Ðe - apply to even indices (1-indexed):
Ḥ   -   multiply by 2                     -> ['a', 'bb', 'c', 'dd']
    - implicit, smashing print            -> abbcdd