It's a very tired argument that gets pulled out and rebutted each time. The Monero blockchain is currently 5.5x the size of the Bitcoin one for comparable total transactions (so linearly larger than Bitcoin's). So when we've had 44 million transactions (as Bitcoin has over its 5.5 year existence) our blockchain will be about 110gb vs. Bitcoin's current 20gb blockchain. This is, in itself, not a problem, as by the time we get there in a few  years disk space will be appreciably larger, and we'll have the same full node problem Bitcoin has (who seriously keeps the full 20gb Bitcoin blockchain on their laptop, for instance) - the majority of our userbase will use lightweight wallets.

A lot of the people that state that Monero has a "blockchain bloat" problem are picking up snippets of conversation between quite intelligent people on the matter without actually understanding the issue. Monero has exactly the same "bloat" problem as XC, DarkCoin, and anything else that uses a form of mixing - you are going to incur additional entries in the blockchain for every mix (or in Monero's case for every additional signature in a ring), which means the blockchain for all of them is going to be linearly larger than Bitcoin's for the same number of transactions. It is a compromise you accept if you want transaction privacy: it uses more space in the blockchain. However, the advantage that a Bitcoin-derived altcoin has is that it can prune the bloated blockchain, whereas with Monero you can never tell if a utxo has actually been spent or just used in a ring signature, so pruning in the Bitcoin sense is not possible. THIS is what they're actually claiming - that all of the blockchains are going to bloat, but Monero's can't be pruned the way Bitcoin's can. It's very, very important to note alongside this that the Bitcoin blockchain has never been pruned, the code to operate off a pruned blockchain is simply not there (that notwithstanding, as of Bitcoin Core 0.9.0 it does have the ability to prune provably unspendable outputs, but that is not the same as the blockchain pruning we are referring to). Therefore, none of these Bitcoin-derived altcoins are actually able to prune their blockchain, despite their belief that they can flick a switch and voila, magically small blockchain. Not unless they have the ability to write code that the Bitcoin core developers and hundreds of contributors have yet to write.