What happens when my accumulated depreciation has reached the same value of my asset?

Once it's deprecated, it has a value of zero, and you can't make any further depreciations.

Let's say it is a ten year old car. You (private person Tix) can buy that car from the company (company Tix corporation) for a dollar, creating one dollar of taxable profit for the company, and privately drive the car until it falls apart. The company could sell the 10 year old car for $2,000 to some third party and have a taxable profit of $2,000. Or the company can just continue using the car, deducting the cost of running the car and repairs, but with no further deprecations.

There are fixed rates for deprecation to make tax reporting easier, especially if otherwise you would have to do subjective things like estimating the value of a six year old car. So the tax office doesn't car what the actual value of the car is, they just car that you make deprecations according to their tables. Whether you look after your car like a hawk and in ten years it is still worth a lot of money, or whether you drive like a maniac and the car is destroyed after 5 years, the deprecation is the same.