You will need at least 3V for most electronic circuits.
Can you use two AA's to power your circuit?

 

a package that would accept two double A batteries, but only used one.

A single https://www.google.com/search?q=carbon+zinc+chloride+AA+voltage may actually be as high as 1.7V when it's brand new ← is enough to power light up a single LED (or a set of LED-s that are connected in parallel) - so they can be seen and in diffused daylight

But it's definitely not enough to drive ?conventional? mosfet-s nor light a led as a collector load of the bipolar transistor or SSR

though they can be lit from boosted voltage but the single AA battery has some 40% of it's rated capacity at heavy drain so ,
say - the 2400mAh (milli Ampere hour) battery would provide 900mA·h at average 1.2V e.g. 1100mW·h
the conversion losses are some 31 to 73 % ← that leaves avg. 1100 · (100% - 52%) ≈ 530mW·h useful energy for LED-s

the low power LED would use 18mA · 3.2V ≈ 58mW → you can power
1 LED for 530mW·h ÷ 58mW ≈ 9h /// 2 LED-s for 4.5h /// 12 LED-s for 45 minutes /// 108 LED-s for 5 minutes (and this is optimistic estimate)