If you're just talking about company expenses, then $10,000 will go a long way. You need $500 to set up a company with Stripe Atlas [1], $100 to purchase the initial shares, $100/yr for a registered agent, and $100/yr to file taxes. You can do your own bookkeeping or use Bench for $135/mo. Stripe Atlas also gives you $5k of free AWS credits, so you don't have to pay for hosting for a few years. Then you can spend the ~$9k on whatever you want (other SaaS services, design, landing pages, etc.)
Apart from the Stripe Atlas fee and share purchase, I didn't put any money into my company. I was still doing contract work and wasn't taking any money out of the company, so it was "profitable" as soon as I had a single paying customer.
I think ~$1,000 is probably the bare minimum if you want to start a company and bootstrap a SaaS product. But it's always better if you can have a healthy bank balance for unexpected things.