Medicare doesn't cover 100% of my medical and dental expenses. Is there a tax advantaged plan I can start to contribute to in order to pay them? I have no employer to 'offer' one.
Tax advantaged plans in the US only reduce tax on earned income. Since you don't have earned income, you don't have that as an option. Instead, what's available to you are normal tax deferred or deferred solutions like a home equity line of credit, annuities (if you have assets generating income - which would be taxed), or selling assets if capital gains tax will be a savings versus other options.
Joe's answer points out a way in which HSAs can be beneficial though as he points out, this is a pretty narrow case which may not apply to the OP.
The only tax-advantaged account that's available for people without earned income is an HSA (for which eligibility is based on having a high-deductible health plan, or HDHP). That shelters the earnings of the HSA itself, if the money sits there long enough to gain value through interest, dividends, or increase in value of a mutual fund or similar.
However, HSAs are not an option if you are enrolled in Medicare part A or part B. As such, you don't have an option for sheltering your medical expenses. You do have the possibility of deducting them after the fact if you itemize and they exceed 10% of your adjusted gross income (AGI), or 7.5% if you are 65 or over (which I assume you are from your references of Medicare and Social Security). If you have quite low actual income, it may well be beneficial to do so (if you are itemizing for other reasons and/or if they exceed the standard deduction). That doesn't require a special account, simply saving receipts and totaling the expense come tax time.