Most of the current growth is due to empty one-minute blocks, which is not inherent in the design at all, and can (and probably should) be changed to a slower block target. For example, as I write this 5 out of the previous 6 blocks are all empty:
If all blocks were empty and the average block size was 350 bytes, that amounts to just 500 kb a day from blocks. If it were 2 min block time, that's 250 kb. That's not a big difference, and I don't think much to worry about.
Very little of the actual current growth is due to the size of ring signatures. Increased usage will cause sub-linear incremental growth because the first thing that will happen is that empty blocks will have transactions added to them (reducing chain overhead from 100% to <100%) and the next thing that will happen is that block headers will be amortized over a larger number of transactions.
In some hypothetical future if XMR has transaction volume that is comparable to BTC, its block chain will be several times larger since its transactions are larger. If that happens, the market will have accepted that a constant factor (very likely <10x) of increased chain size is an acceptable trade off for increased privacy of transactions on the chain. I see no way to objectively predict whether that will happen.
The current daemon softfork limits the blocksize to ~192 KB to avoid propagation issues, which may be exacerbated when we move from in-memory DB to an actual database. Of course, miners may make larger blocks if the median allows for it, but it's very likely they'll be orphaned.
This gives us about 2x the theoretical bandwidth of Bitcoin. End of the day, I think what will happen is that fees will begin to dictate tx inclusion, and that tx via Monero will probably be more expensive than for Bitcoin (given unlikely equal adoption).
Softfork to mandatory minimum mixin of 3 will bloat the chain about 3-fold further, but as it currently stands this is not a huge issue.
Most computers cannot use XMR without a database, because the chain doesn't fit in memory. I'd consider that urgent. Not urgent enough to do it wrong, however.
I'm working with tewinget right now on the database, so it won't be that far off in the future.