unix - Unlimited Bash History - Stack Overflow
http://*.com/questions/9457233/unlimited-bash-history

Set HISTSIZE and HISTFILESIZE to an empty string:
HISTSIZE= HISTFILESIZE=

In bash 4.3 and later you can also use HISTSIZE=-1 HISTFILESIZE=-1:

n.  Setting HISTSIZE to a value less than zero causes the history list to be

    unlimited (setting it 0 zero disables the history list).

o.  Setting HISTFILESIZE to a value less than zero causes the history file size

    to be unlimited (setting it to 0 causes the history file to be truncated

    to zero size).