Mind that typical camera sensors work in a cumulative/integrating way - think of either a tiny capacitor either being charged by a voltage source with a tiny photocell (photodiode, phototransistor, CDS cell, whatever!) in series, or a pre-charged capacitor shorted by a photocell. IIRC the second way is actually what is happening in typical sensors.
This structure will stop integrating correctly when the capacitor is empty (or charged to the available voltage). The meter needle is pinned, so to speak.
On-chip capacitors are relatively large, so you cannot make them arbitrarily high in capacitance; also, you would end up with smaller voltage differences for the same amount of light integrated, and thus with more readout noise.
A typical sensor pixel is not a non-integrating photocell with a buffer and a sample and hold circuit (activated at readout) attached - such an arrangement would likely be much noisier and would also sabotage the meaningful usage of a traditional shutter. Also, would be much more complex (see below).
One might be able to put some kind of logarhithmic amplifier between a low capacitance photocell and the actual capacitor ... complicating the circuit much. Also, you'd need to shield any non-trivial analog circuitry in the sensor pixel from light very thoroughly: Silicon semiconductors are both somewhat translucent to IR, some of the structures used in making especially MOS/CMOS parts are actually quartz (which is translucent to a lot more), and all that stuff is photosensitive even if not intended as a photodiode or phototransistor. Log amplifiers especially are very dependent on circuit balance, thermal or optical interference is very good at upsetting it...
A path that maybe hasn't been explored much yet would be to use a different filter structure on top - for example, an R, G, B, R+ND8, G+ND8, B+ND8 mosaic... Blooming might ruin the fun, though.