I have a pair of tall upper cabinet on each of our his and hers vanities. I love them for tall products and hair appliances but that probably isn't the look that you're after.

I've always liked this idea I found on Houzz; maybe it would work for your space.

[contemporary bathroom design[(https://www.houzz.com/photos/contemporary-bathroom-ideas-phbr1-bp~t_712~s_2103) by san francisco interior designer Olson Design and Construction

We have separate vanities. Mine is a rather long counter with a tall 3-door cabinet built in the corner that I use for storage of tall items, extra towels, and with a pull-out laundry basket behind the bottom door. Love it but it may not work for your sleek Asian look. The idea above is cool too!

Take a shelf out of the medicine cabinet to make room for tall items.

A really deep drawer would also work, or an undersink pullout shelf.

We have tall 15" wide linen towers flanking our vanity for the tall items. Actually, we use them for just about everything other than hairbrushes, tooth brushes/tooth paste & nail files/clippers. Storing vertically on shelves just works better for us. Mom has a deep drawer in her vanity. Its the depth of 2 framed drawers with a faux front to make it look like there are really 2 drawers there. Its tall enough to handle shampoo bottles, hairspray, bubble bath, etc. I really liked her setup & put it on the vanity order, but our cabinet guy messed up the order and I didn't get it. The linen towers work best IMO, followed by the deep drawer.

Hope this helps!

I keep most of my daily-use products in my medicine cabinet, and arranged the shelves to allow for taller and shorter bottles/containers. A revamped package (of make-up remover) is taller than its predecessor, and would not fit on the "designated" shelf along with its cap. Since the cap isn't really necessary for storage (pump bottle), I removed it so it would fit on the same shelf (I like keeping similar-use products together--I'm OCD that way!).

A new (to me) hair product, in a too-tall for any shelf bottle, one that wouldn't be used every day anyway, is stored in the vanity (in the open section behind sliding doors). I also store my big bottle of aloe gel, tall tube of body cream, hand creams, back-up bottles of shampoo, conditioners, soap, and bathroom cleaning products, etc. (again, all in their "designated" sections/spaces) in the open section under the sink (behind doors). Spare rolls of tp, box of tissues, blow dryer (rarely used), and other assorted extras, are stored in the two drawers below the sliding door section.

We are just finishing up, so I haven't used this yet to tell if I like it or not, but I chose to do a 'spice rack' pullout for tall narrow items like hair spray and lotion.

Link attached for rev-a-shelf pullout.

Here is a link that might be useful: http://www.rev-a-shelf.com/p-4-cabinet-pullout-organizer-with-wood-adjustable-shelves-and-bins-for-bathroom-vanity.aspx

I think that spice rack idea is terrific. I've even seen a picture somewhere of an even larger and taller pull-out rack that is built into a wall in the bathroom, such as a narrow wall between vanity and toilet.

So far I have managed OK with standing items in a deep drawer in my vanity. Good luck!

We (my husband) built a custom cabinet for our bath rehab. If you are gutting walls, you have the option of building it as tall as you want, and deciding ahead of time to have shelves of varying height. We're having a mirrored front, because I love mirrors. But you could adapt the design for your specific plan.

Before we decided to do our own, I was particularly enamored with Roburn. They make a 6' tall, crazy big cabinet that I was in love with. It's not an Asian style, but they make the same design without any trim that can be completely flushmounted. The interior is fantastic. All adjustable, gorgeous interior mirror, basket, lighting... pretty much the greatest thing ever.

We have shallow tall cabinets on each vanity countertop, with adjustable shelves inside.There's also cabinet space down below.

And then in the shower, we have a multi-shelf niche that can handle tall bottles.

LOVE that full length mirrored built in cabinet!!!
What a cool idea.
I keep my tall items under the sink with a pull out shelf.

Well, I keep my tall stuff in a plastic box under the sink...

My concern was with the split sinks, where to put stuff we share like mouthwash and the water pik, so we built an "appliance garage" in the middle stack with an outlet inside.

However, you could do something similar designed for tall storage that is out of sight. And given how you finish it, it could look very asian.

We decided on a center stack. I went to the cabinet builder asking about side medicine cabinets and he was not a fan. He talked us into the center stack. And my husband even gets tv space at the top (2 against 1 at that point). So much for the minimalist asian-inspired theme!

Thanks for all the suggestions folks.