That's basically what an HSA is.
You sign up for a high deductible plan where you pay for your own medical expenses, but document them, up until you hit your out of pocket maximum ($8300 for individuals or $16600 for families), at which point your insurance kicks in to cover the catastrophic bills you typically won't have in a typical year.
Meanwhile, you are eligible to contribute $4300 per year for individuals or $8750 for families into an HSA, which has very favorable tax treatment (pretax money deposited, not taxed when taken out for health expenses, even after growing a lot), and allows you to invest everything above the minimum cash balance (varies by provider, usually something like $1000 or $2000).
That way in a year you happen to hit a $1 million illness or injury you're still covered against catastrophic financial loss, but you generally pay your own way with tax-deductible funds that you're allowed to invest for growth.