I don't know about TX law but CO does not require a business owner to carry Workman's Comp.
All states have WC laws. The minimum number of employees will vary by state.
Here in GA it is 3.
You count W2 employees, not 1099.
Even if you do not carry WC that does not exclude the employer from liability. If I work for you and I am injured on the job I can still come after you for med payments, LOT and other issues.
If you meet the minimum threshold (3 employees for example) and do not have WC you must purchase a bond to cover your liability. Many large employers will either post a bond or pledge assets and purchase a cover (reinsurance) to take care of claims in excess of the bond.