It may not be an ideal option, but you could use an HSA as a tax-sheltered investment vehicle.
The contribution limit is only $3,350 for an individual and $6,650 for a family in 2015 (plus $1K if you're 55+), so you're only making up a small portion of the 401(k) limit. Also, you (and a family member to get the higher contribution limit) have to be covered by a qualifying high-deductible health insurance plan (HDHP) to be eligible to make HSA contributions. As such, it may not be the best option if you regularly incur significant medical expenses. And in many cases, the investment fees in an HSA are higher than you would find in a 401(k) or IRA. The investment choices can be limited, so it is important to research the options before selecting a provider.
All that being said, the contributions and growth are both tax-deferred (tax-free if you use it for healthcare). Then at age 65 or Medicare eligibility you can withdraw the funds without penalty and pay only income tax, even if they are not used toward healthcare expenses.