Which point-and-shoot digital camera will have the biggest lens?

First, a little reality check. Cameras like the Nikon P900 and P1000 which have enormously huge zoom ranges (83x and 125x respectively) are very popular because of their zoom ranges and NOT because of their image quality. These zoom ranges are cool on paper but in reality these cameras are difficult to use and produce less than stellar results BECAUSE of their super zoom ranges. Furthermore, unless you're photographing wildlife or sports like a surfer, you simply won't come across situations where these super-high focal lengths are actually needed. Most pros will go their entire careers without ever using a focal length longer than 300-400mm. Sure, it's cool to take shots of the Moon, but after you do that a few times, then what? You end up with a camera that has a very complicated, and therefore inferior, optical quality. The longer the zoom, the worse the optical quality.

Another aspect of these cameras that result in poor image quality is their sensor size. All of these cameras have to have very small 1/2.7" sensors. The smaller the sensor, the smaller the lens. This is how manufacturers can squeeze an 83x zoom into a camera that fits in the palm of your hand.

Unless you have a need for such huge zoom ranges, there's no reason to buy a camera with more than a 10x zoom.

Within your price range, the camera with the best image quality will be an DSLR like the Nikon D3400. Okay, that's too big. There's an answer to the size issue which is the Sony A5100 which sells for $350 body only. Both the D3400 and A5100 use a large APS-C sized sensor and have interchangeable lenses which will allow you to greatly increase your image quality and creativity as your budget allows. You won't get super-high zooms with these cameras. The strongest zooms are lenses in the 28-300mm range or about 10x zoom. If you need a stronger telephoto, then you can add a 1.4x or a 2x teleconverter.

If you want a pocket camera, then the best one under $400 is the Sony RX100; nothing else really competes with it in terms of usability and image quality. Nothing can, because nothing else in this price range uses a 1" sensor like the RX100.

To compare image quality of cameras with fixed, non-interchangeable, lenses go to https://www.dpreview.com/reviews/image-comparison. You can't use this tool to compare image quality of interchangeable lens cameras like a DSLR or mirrorless because the lenses used vary, and it's the lens that determines image quality much more than the number of pixels in the sensor. This is why a cheap camera with a great lens will always produce a better image than an expensive camera with a cheap lens.

Thoughts:

1. In photography, you get exactly what you pay for.

2. Good zoom and good optical quality never go together.

3. If you find dSLRs bulky, look at mirrorless cameras.

Define "biggest lens". Is that by lens diameter or length?

For Point and Shoot, anything from Canon, Nikon, Sony, Pentax, and Olympus in your price range will fit. The lens sizes will be similar as will be the imaging chip sizes. You may be able to get into a "bridge camera" (looks like a dSLR; has a fixed - not interchangeable - lens) which will have bigger optical zoom.

Mirrorless cameras are taking over from Digital SLRs

They have just as good picture quality as a DSLR, with similar functionality and can be used as a simple point and shoot. You can also get other lenses for them.

I currently use an Olympus OMD - it's pretty damned good and not that expensive