When I record a billable expense of $100 with HST on it$13, I want to bill my client only$100. However,with the billable expense feature,it invoices for the full $113.
You've come to the right place for help. I'd be happy to share more info on creating Billable Expenses in QBO.
Based on the info you've provided, it sounds like you don't want the HST to be included. If this is the case, I recommend removing the tax option when creating the expense. I've attached a screenshot below which shows you the two spots to remove the HST when creating an expense:
Let me know if you have other questions. If not, I suggest checking out this helpful article which explains more about creating billable expense: Enter billable expenses. I'll be here in case you need further assistance.