This said, the blockchain will grow larger than bitcoin one, by a linear factor. Price to "pay" for unlinkability and untraceability.
For now. I'm pretty sure there will be a much smaller chain with adapted cryptonote before too long.
Chain is going to be larger for this one, probably some 5-20x larger than that of Bitcoin's. That being said, that means maybe hundreds of GB at worst in a few years, which isn't an impossible tradeoff (for now) with privacy.
There are a lot of propositions for means to reduce the size of the blockchain, and we're reviewing them all carefully.