Is it possible for computing power to increase quick enough to break mining retargeting? So from what I understand,  mining bitcoins basically becomes more difficult as more bitcoins are mined.  This is done so computing power doesn't get so strong that bitcoin mining becomes trivial to the point where people could mine 100's of bitcoins a day using their cell phone, for example.

Assuming that my understanding is correct,  do any of you math wizards out there know if there will ever come a time when the computing power is strong enough to make the mining time insignificant, or will the blockchain always be ahead so to speak because it becomes more complex as more bitcoins are found/rewarded/whatever? 

Very curious about this.  Thank you for any answers.
The difficulty of the mining algorithm is solely reliant on the time it takes to mine 2016 blocks and the previous difficulty. (See: How is difficulty calculated?). 

As the mining algorithm retargets exponentially (up to a factor of 4) every 2016 blocks, it is impossible for technological progress to outpace difficulty increase in the long run. Even if the mining power were to double every day (which is insane!), the block interval would only go down to about every two minutes in average, i.e. approximately five times faster.

On another answer I've provided a sample calculation for an overnight hash power increase by factor 21, which you might find interesting. The difficulty would catch up in 14 days.