Yeah in theory you could start for free.... once you get into paid traffic though you need enough money to test multiple different things to find where you can improve. It's hard to give a number because there are sooo many factors to consider.
For example, You could be running a high ticket $1000 CPA offer on Linked in Ads and perhaps it is in the investing niche - you would need thousands of dollars to test this.
or maybe you get paid $12 per lead and the offer has an epc of $2 in the finance niche, and you found a source of cheap traffic on google search....here you only need less than a couple hundred to test...