Is there a maximum balance that a single address can hold? Is there a maximum number of satoshis that a single address can control at a single time? I ask because when I generate regtest blocks to a single address, it's balance gets stuck at 1494999998350 satoshis (14949.99998350 BTC), and doesn't increase anymore from that point. All balance for that address is from coinbases, not vouts.
Nope. Not 1 bit. A balance (in this context) is the sum of all UTXO's for a given address. There is no limit on the number of UTXO's or their total amount (from the perspective of how the blockchain works and the blockchains limitations). 

Software interpreting the balance may have limitations on consuming, processing and/or displaying a number beyond a point. Int32 vs Int64 for example: https://stackoverflow.com/questions/9696660/what-is-the-difference-between-int-int16-int32-and-int64

Here's an address with a balance of 138,660 BTC at the time of this post:
https://longhorn.bullpay.com/?gateway=2&amp;data=3D2oetdNuZUqQHPJmcMDDHYoqkyNVsFk9r

Note: This answer is specific to the question and specific to Bitcoin. For the sake of brevity situations where "creating" tokens or currency on another chain in amounts that could theoretically "break" anything is not covered.