I used this variant:

alias gitb="git branch 2> /dev/null | sed -e '/^[^*]/d' -e 's/* \(.*\)/\1/' | pbcopy"

just for anyone using bash for Windows, here's their equivalent command:
git branch | grep '^\*' | cut -d' ' -f2 | tr -d '\n' | clip
Bash for windows has some but not all of the Linux commands installed, and pbcopy is one of those which it doesn't have.

As of Git 2.22 just:

git branch --show-current | pbcopy

Looks like that last one still needs the newline removed

git branch --show-current | tr -d '\n' | pbcopy

@smithtimmytim that one removed any "n" chars from my copied branch name.

@smithtimmytim that one removed any "n" chars from my copied branch name.

@MSPigl You probably did tr -d 'n' instead of tr -d '\n'. (Probably he updated his comment fixing that 😅 )

for those who end up here after today, there is a command built into git to show the current branch

git branch --show-current

Add these get aliases to your .gitconfig and you have the same functionality

[alias]
	scb = branch --show-current
        ccb = ! git scb | tr -d '\n' | pbcopy

git scb - shows (prints) current branch
git ccb - copies the current branch to clipboard