Here's a really good video describing several different ways to magnify sunlight to start a fire.
Don't dismiss this as not helpful though -- it gives more than one way to improvise a magnifying 'apparatus'
I'll try and describe with words, but the visual example is linked below.
Take a look, sound isn't necessary if you want to mute the video. It's aboutAbout 3:30 long: https://www.youtube.com/watch?v=HCyHC7lnMyQ
Get an old light bulb, the kind with the filament. Break into the bottom of it (the part that spins into the socket) with a screw driver or some such. Be careful to keep the glass intact. After getting the filament out, use a pinch or 2 of salt to clean out the white powder. Swish/swirl it around inside and it'll clean the powder off. Rinse, then refill and you have yourself a curved lens for magnifying.
Get a small bowl/candy dish, the inside should be about the size of your fist. About 2 feet by 2 feet of clear plastic wrap, center the wrap over the bowl, then line the inside of the bowl. Fill the bowl with water about 3/4 full. Then bring the corners of the plastic wrap together, gently lift the water out of the bowl. The water will make the plastic "sag," so carefully twist the top of the plastic so it ends up like a small baggy. Keep twisting until the baggy becomes tight like a ball. Presto! You have yourself a curved lens for magnifying.
Less practical, but it'd work. Take a picture frame without the glass or backing or anything, just the frame. Place it on top of two stands, as if the frame was about to become the top of a table and the stands were the legs of that table. Grab that handy plastic wrap and spread it across the frame nice and smooth. Tape the plastic securely to the frame. This would be the clear, plastic wrap, table top. Slowly pour water onto the plastic and it will sag, creating a lens. Put whatever you want to magnify under the table and look through the water to see it magnified.
Using a water bottle, with water in it. Near the bottom of the bottle or the top of the bottle, you should be able to angle it just right to get a magnified image.
The video uses water and a chemical reaction to create heat, which doesn't address the question at all.
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My own thoughts:
A whiskey glass with just the right amount of water in it. Whiskey because the bottom is already in a concave shape, so the water just increases the effect (I'd imagine)
A cell phone camera with zoom and a steady hand.
Certain flashlights have a magnifying lens in the front to help diffuse/focus the beam of light. You could try taking that off and looking through it.