I'm doing a little bit of experimentation with a glass of water and a 9 V battery. I added a little bit of salt to the water, and I measure the resistance in the water from probe to probe to be ~60 kΩ. Ohm's law tells me the current (given negligible additional resistance in wires, I'm using short copper wires) should be ~150 µA if I connect the terminals of the battery through electrodes separated in the water. When I drop the wires in there I can visibly see small bubbles forming on the electrodes, which tells me that current is indeed flowing, and chlorine and hydrogen gas are being formed.
However, when I then proceed to detach one wire from the battery and connect it to one of the probes of the multimeter, and then connect the other probe to the battery instead (connecting the multimeter in series with the circuit, as you're supposed to to measure current), then nothing happens. The multimeter is supposed to be able to measure to a resolution of 10 µA, so ~150 µA should definitely be within its detectable range. However, it's not just that there's no measurement, but there are no bubbles forming, and thus presumably no current (or at least not nearly as much). I've connected the probes together measuring resistance to ensure that no fuse has blown, and it's measuring only ~5 Ω, so there's clearly a connection, and this resistance should be negligible compared to the ~60 kΩ in the water.
So, why isn't there any detectable current flowing when I'm doing this?
Here's a schematic, as requested by several commenters:
On top is the connection with just the battery and wires in the water, below is the multimeter connected in series with that circuit.
The multimeter is a Biltema 15-133.