Life expectancy of an electronic shutter

No an electronic shutter doesn't have a limited life as far as shutters go. It does have a life span as far as any electronics goes.

A dSLR can have either a 100% electronic or a mixed mechanical and electronic shutter. Indeed Sony Alpha have both pure electronic (called silent shutter) and a mix called electronic first curtain shutter.

Naturally the traditional 100% mechanics are still the dominant form based on IQ: electronic first curtain shutters show virtually no reduction in IQ, however 100% electronic shutters still show some artifacts in certain situations - however it's mostly due to the CMOS sensors - if using CCD sensors then the artifacts are virtually gone. However CCDs have their own drawbacks (like high ISO noise).

Cameras that have both electronic and mechanical shutters tend to leave the mechanical shutter closed when the camera is turned off. This means that in order to use the electronic shutter the mechanical shutter must be opened at least once each session. At the end of the session the mechanical shutter is closed and that will finish one complete cycle of the mechanical shutter.

When deciding whether to use mechanical or electronic shutter there are other things besides wear on the mechanical shutter to consider.

First and foremost would be the effect of that choice on the quality of the image produced. Sometimes a mechanical shutter is more useful to produce the image desired. Sometimes an electronic shutter is more useful. Sometimes other considerations, such as the amount of noise a camera makes and when it makes it, are also in the mix.

Second is the effect on the sensor of leaving the shutter open for long periods of time. This tends to cause the imaging sensor to heat up. As image sensors heat up they tend to produce more noise and the overall signal to noise ratio degrades. The images produced become noisier for the same scene shot with the same settings. When professional television production companies first started using DSLRs to shoot scenes for television shows they could only use a particular camera body for about ten minutes or so until the sensor got too hot and the image got too noisy (and this is with HD video, which is only 1080P and thus enjoys the advantage of noise reduction as a high resolution image is reduced in size)! They would then have to switch to another "fresh" body while the first was allowed to cool for up to an hour before it could be used again. Newer high end DSLRs tend to have better sensor cooling than those early original 5D and 7D bodies, but heat build up is still a concern when using them for professional productions.

Third is the eventual result of the second consideration: the more light a sensor is exposed to for a longer period of time the greater the number of pixels that develop defects. The number of stuck, hot, or dead pixels increase over the life of any sensor. Leaving the mechanical shutter open and the sensor energized and exposed to light over the course of an entire shooting session will reduce the life expectancy of the sensor by a considerable amount compared to using a mechanical shutter and only energizing and exposing the sensor to light when actually recording an image. If you think replacing a mechanical shutter on a digital camera is expensive, you don't want to know what it will cost to replace the sensor!