Is There A Device For Home Use That Can Measure How Low The Bass In My Speakers Is?

For any measurement to be meaningful, you will need a pink noise source and a precision sound pressure level meter with a calibrated microphone.  You might be able to download a 1/3 octave pink noise source which you can burn to a CD.  Records or prerecorded tapes will of course be influenced by the frequency response of your cartridge and tape machine, which at very low frequencies could add another variable.  Where you place the SPL meter and what weighting filter you use will also affect your results.  Are you trying to measure just the speakers or the total response of your speakers in your listening environment?  Bruel & Kjaer wrote much on acoustic measurement and have published many books and booklets on making meaningful SPL measurements which I would highly recommend as useful reading.  Even the instruction book for their 2203 Precision Sound Level Meter has useful information for doing what you want to accomplish.

Listen to Chayro,  The Stereophile test CD will tell you.  Common sense tells you when it becomes inaudible, you have passed the bottom end.

If this is just for curiosity sake, a test tone generator is on many websites including audiocheck as listed above.  Also there are phone apps for SPL, so you can verify the drop in dB, probably not super accurate, but good enough for curiosity. 

@mitch4t 

hypoman is the first, and only, contributor to this thread to even mention an essential parameter relevant to almost every other suggestion: that the SPL meter must be properly "weighted." The meter elliotbnewcombjr actually provides a link to will be useless for measuring bass output: it is "A weighted," which does not register anything below 100 Hz. The vast majority of such inexpensive meters available on Amazon or eBay are "A weighted," like the one in that link, which are intended for measuring industrial noise levels, not music. I learned this the hard way: I acquired one of those "A weighted" meters, and was puzzled that the readings I got seemed impossibly low for the subjective loudness they presumably measured. A device that will allow you to switch between "A" and "C" weighting will dramatically reveal the difference. And since the OP is interested specifically in "how low" his speakers will go, none of the suggested meters in this thread will be of any use at all in determining this.