Multiple encodings with any lossy format will result in additional quality loss. This is what is called generations of loss and is a concept that dates back to analog techniques where each time you made a copy, the quality of the copy was inferior to that of the original.
Digital tools allows us to avoid generational quality loss when using lossless formats (formats that store all available information without deterioration), however lossless formats require much more space and are therefore rarely used for final output
Lossy formats such as JPEG work by storing an image that closely resembles the original but throw out information that they deem to be unnecessary. Encoding an image more than once will generally reduce quality further since the algorithm does not have the original full quality image to work from and must instead encode the lower quality image.
The file size will generally give a rough estimate of loss of detail if you are encoding within the same compression scheme, at least when files sizes decrease. It is however possible to increase the size of an image (say for example you had a lossless image, saved it as a 10 quality JPEG and then resaved that as a 100 quality JPEG). While the file size would grow, it would be storing noise at high quality rather than information from the original image.
In theory a tool could compare the pixel data of an original image with the differences in the final image to get a better estimate of information loss, but it's hard to quantify how meaningful those changes may be. For example, if every pixel in the image was a few color values brighter, the overall image would seem very similar, where as if one pixel was off by a large value, it would be much more obvious, even if the entire rest of the image was untouched. Similarly, if areas of fairly uniform color are actually made uniform, it may be a large change in pixel values, but have a limited impact on how someone perceives the image.