Is the International System of Units complete? Are there any known (measurable continuous) physical quantities, which are neither base quantities of SI, nor are derivable from the base quantities?
In other words, are there any quantities which require units not part of SI?
I am not including in this question quantities that are dimensionless (such as angles), or that have an inherent unit (such as discrete quantities, e.g. 1 bit). 
 A: As you get farther and farther from electro-mechanical physics, you'll pick up more and more non-SI units that are arguably still "physical".
For example, the CIE system of color has measurable continuous values that aren't, strictly speaking, derivable from SI base units.  

(WikiCommons, see above link). The axes are "X and Y chromaticity".
Further away, getting into the social sciences, you'll find things like Gross Domestic Product that are measurable and continuous (at least in the sense the electrical charge is), but not directly related to SI units. 
Are those physical?  Yes, in the same sense that potential energy is: They are caused by, and can in turn make stuff happen in, the physical world.
A: No.
Information is not a quantity which has a unit given in the SI that is arguably of fundamental physical importance and a universal quantity (i.e. is applicable across the entire Universe, as far as we can understand). Granted, it could be considered dimensionless, but then again, so could plane angle and the SI provides the radian to measure that. The units bit (or "shannons" when referring to information had, not information storage capacity), byte (= 8 bits), nat (~1.44 b), hartley (~3.32 b), etc. are not SI units.
