You have a very tiny miniature car (much smaller than an inch, consider it a point) on a circle that is 100 inches in circumference (i.e. the length around the circle is 100 inches). At the beginning, the car is at a point on top of the circle, say at 12 o'clock at time 0.

The car starts moving at uniform speed, clockwise around the circle, and goes 1 inch in 1 second as its speed. Also after each second, the circle is stretched, so that 100 inches are added to the circumference after each second. As in the radius increases in whatever amount needed such that 100 extra inches are added to the circumference after each second.

So picture it as follows: at t=0, the car is at the starting point at the top of the circle. The car travels 1 inch by the time t=1s, and exactly at t=1s the circular track expands by 100in instantaneously, then the car travels another 1 inch along the circle by the time of t=2s, at which point the circle expands yet again so 100in are added again to its circumference, and keeps going in this fashion.

Does the car ever make it around the whole circle, back to where it started at 12 o'clock? If so, how long would it take?

Sounds like no.

i came up with 350

Surprising as it is.....the car actuallymakes it around the whole circle.