My first opinion on this question is … really? If you can't afford between $10-$20/mo on a decent email automation tool, then you're doing something wrong.
Anyway, many email automation tools offer either some sort of trial period OR a free tier.
ActiveCampaign has a 14 day trial.
iContact has a 30 day trial.
GetResponse has a 30 day trial.
Constant Contact has a 60 day trial.
MailChimp offers a free tier; however, it does not allow for automation – so, it will not suite your needs. It's great if all you need to do is send broadcast emails.
As for totally free autoresponder services – they're more or less crap. Tracking, automation tools, features, ease of use, deliverability rates, all that stuff … is hit or miss … and I would never recommend a free solution for your business.
Spending $10-$20/mo on a quality autoresponder service is money well spent. Instead of drinking Starbucks coffee for 4 days, brew your own – and that'll more than cover the cost of an email automation tool.