No, it doesn't work that way. The image file isn't built up as the exposure goes on, but rather is made from a full read of the sensor when the exposure is complete. So, you're not writing more data to the memory card when you expose for a longer time.
Each photosite — one "pixel" on the sensor — is a counter that goes up as it's hit by more photons. It's actually an analog device, but when read at the end of exposure, a single digital value is produced (usually 12 or 14 bits). This value is simply the total amount of light that site received. If the particular pixel is all dark, it'll be
000000000000, and if it's all light, it'll be
111111111111. There's no record of how long it took that all-full sensor to get to that state — it could be very, very bright with a wide aperture, so you could get that value in ¹⁄₁₀₀₀th of a second. Or, it could be so dark out that it takes 30 minutes to get the same result.
In the end, though, it's the same single value. And all together, there's no more or fewer values no matter how long the exposure is.
There is another factor, though. Some files will compress better than others. Patterns compress well, and of course large identical areas compress best of all. Arbitrary detail compresses poorly, and random data worst of all. Since noise is by definition random, very noisy images produce the largest files. There isn't a direct correlation to exposure length here, but longer exposures may have more noise as the sensor heats up. So, that may be a practical consideration, but it isn't because of the accumulation of data per se.