Python script stopping after a while

Let's not worry about getting it to run every time your computer starts up until you can get it to run continuously in a normal login session.

If you're sshing into the computer and starting it, just leave the ssh session open, and watch what it prints out. If you can't do that for whatever reason, capture the output by doing, e.g., python foo.py >foo.out 2>foo.err instead of just python.py. Either way, you'll be able to see why it's stopped, instead of just knowing that it's stopped.

You should also look at the log.log file that you explicitly create in your code, and at the syslogs (/var/log/syslog by default on most linux distros), to see if there's anything relevant there.

Also, please tell us exactly how you're "run[ning] the script from ssh". If you're backgrounding it with &, sending it as the ssh command, etc., the instructions above need to be modified. (But you don't have to wait for the modified version—just this once, ssh into a normal login script, run it without backgrounding, and leave it running.)

With only the information that you've given us so far, it's impossible to know exactly what's going on. But my first guess is that you're getting an unhandled exception, which you'd see as an exception traceback, something like this:

Traceback (most recent call last):
  File "exc.py", line 7, in <module>
    foo()
  File "exc.py", line 5, in foo
    print(1/x)
ZeroDivisionError: integer division or modulo by zero

The fix is to handle it. The most trivial fix is something like this:

while True:
    try:
        cam.start()
        img = cam.get_image()
        pygame.image.save(img,"current.jpeg")
        cam.stop()
        host = ftputil.FTPHost(**)
        host.upload("./current.jpeg", "/domains/*/public_html/webcam.jpg", mode='b')
        host.close()
        if !count:
                host = ftputil.FTPHost(**)
                filename = str(time.time()) + ".jpg"
                host.upload("./current.jpeg", "/webcamarchive/"+filename, mode='b')
                host.close()
                count = 10
                logging.info(str(time.time())+": Still running")
        count -= 1
    except Exception as e:
        logging.error('Caught exception: ' + str(e))
    time.sleep(3)

However, you're probably better off looking over each kind of exception you could run into—or looking at the exception you actually are running into—and putting the appropriate error handling and logging in each place, and only using this as a last-ditch fallback.

Also, you should really use with clauses instead of manual close calls. For example, if host.upload() raises, the host.close() will never get called. So, instead of this:

host = ftputil.FTPHost(**)
host.upload("./current.jpeg", "/domains/*/public_html/webcam.jpg", mode='b')
host.close()

Do this:

with contextlib.closing(ftputil.FTPHost(**)) as host:
    host.upload("./current.jpeg", "/domains/*/public_html/webcam.jpg", mode='b')

(Many types don't even need the closing, because they're already inherently context managers, so try that first—but if you get an error about ftputil.FTPHost not having an __enter__ or __exit__, this is how you can deal with it.)