Been thinking about this one well before my War On Plastic post last year, I just couldn’t crack the design until now. I wanted a deodorant stick that can use refills that were similar to bars of soap.
The deodorant bar has grooves that align with internal gears (below left) that freely spin, unless the turn key is locked in place. Bars are easily dropped in and then locked in place, and like a normal deodorant stick they can be adjusted vertically as the bar erodes from use.
To ensure the entire deodorant bar is used, I added LEGO studs so that the leftovers of the previous bar can be plugged into the new bar (above right).
A normal cap can be used for this design, but my preference is a weighted stand (top image on the left) so the stick can be easily parked when not in use — no more dealing with a pop-off cap.
The stick (and weighted stand) would be made from durable materials, so it could last years and years. The refill bars could be sold in cardboard boxes.
As for existing products, both roll-on and gel deodorants could be modified to be refillable tomorrow if companies really wanted.
Dave's Geeky Ideas 
I love the idea of refillable Deodorant sticks. The Lego notch is a stroke of genius. Would the material of the bar be able to hold of to the pressure applied during application to a person’s underarms? Most bars I’ve used in the past are very soft, and I think the pressure would scrape the the grooves off.
The very first bar would see the top LEGO grooves worn off, subsequent bars would be fused with the remnant of the previous bar, if that makes sense. Hopefully the conjoined bars would erode together, and without waste. In theory anyways!