What is the maximum number of blocks in a blockchain?
Actually, there is no maximum limit on number of blocks, blocks just keep getting added to the end of the chain at an average rate of one every 10 minute this process will not stop even if all Bitcoin (21 million) get mined, because people will need blocks to store their daily transactions.
Vijesh's answer is correct at the blockchain level. Within smart contracts, block numbers are represented as a 256-bit unsigned integer, so the maximum block number is 115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,639,935. This would take 3,671,743,063,080,802,746,815,416,825,491,118,336,290,905,145,409,708,398,004,109,081,935,347 years to reach at 15 second block times.
This is unlikely to be a problem in practice.