Any half decent modern wallet will be able to provide you with essentially as many addresses as you want. I can't necessarily say the same for all wallets from back in 2014. Do you know the name of the wallet you were using back then? Are you still using it?
Most wallets these days which are not Bitcoin Core are what is known as hierarchical deterministic wallets. They provide you with a seed phrase, also known as a recovery phrase or mnemonic phrase, which is a list of usually 12 or 24 words. These words back up your entire wallet, and addresses are generated deterministically from these words. By that we mean that the same words will always reproduce the same addresses in the same order. So if you write down your 12 or 24 words, then that effectively backs up every address in your wallet, including ones you haven't generated yet. Theoretically, one set of words could generate trillions of trillions of trillions of addresses, although obviously no one would ever need that many.
Bitcoin Core does not use seed phrases to generate addresses, and so if you are using Bitcoin Core instead of backing up your seed phrase you need to back up your wallet.dat file.
It was MultiBit classic (it was probably just MultiBit back then). I'm not using it as it's been abandoned, I've imported my private keys into electrum, and I'm still trying to consolidate all my inputs in a new SegWit wallet (I'm in no hurry though, but with the current prices it would cost me the equivalent of $150). Sorry for the late reply!
