What will happen when the blockchain size will reach 1 Tb? Or 10 Tb? My question is more about bitcoin client (bitcoind). Will it still be working fine or maybe something will go wrong? I mean, maybe LevelDB can't handle this amount of data correctly?
The database itself does not hold the blockchain. LevelDB is really only used to index where blocks are stored on disk, to store the UTXO set, and to store the current chainstate. So if the blockchain were to be terabytes in size, it should be fine.

The problem would actually be downloading the blockchain and verifying all of that data within a reasonable amount of time.