You could buy a bridge camera. A bridge camera stands between a DSLR and a compact camera, most of the time equipped with a large zoom lens, even up to 3000mm! An example is the Nikon COOLPIX P1000 (price range between $800 and $1000). But let's be honest if you want quality photos, forget these types. We are going to talk only about DSLRs and Mirrorless system cameras.
Nikon, Canon, Sony, Fujifilm, in fact, all DSLR/mirrorless camera brands take beautiful photos.
If you are going to buy a camera, choose your brand wisely and stick to it because you will certainly buy extra lenses; and If after a while you decide to go for another brand, the whole replacement of the lenses can get very expensive (incompatibility).
Size-wise, with sensors, we have two formats to choose from, FX (Full Frame) or DX (APS-C). Most hobby photographers use the DX-format, mainly because of the price tag (although this difference is fading in 2019).
In photography, beginners underestimate the power of the lens! Most of the time, they have a kit-lens that came together with the camera. They don't realise that it is often a cheap lens put in the box to keep the costs down.
For me, the most important element of the camera is the lens. It's the part of the camera where everything starts, if you have a cheap/bad lens, you will never get good pictures, no matter how good your camera is. Particularly for small animals, you will need to invest in a good lens. A good telephoto lens can cost 2, even 10 times more than the camera. A focal length of a minimum of 200mm is required to shoot close-ups of small animals, but then you need to stand close to the subject! A better focal length is 400mm or even 600mm. The lesson is, more is better but also more expensive.
I'm going to use my basic configuration as an example to calculate the price. I have a DX (APS-C) mirrorless camera, the Fujifilm X-T3, together with the Fujinon 100-400mm lens. At the moment of writing, this combo costs around $3500 (€3000).
That's not cheap, but if you compare it with what most wildlife photographers use, it's not that bad. According to some statistics that I found online, the most used and most favourable wildlife combo is the DX (APS-C) DSLR Nikon D500 with the Nikkor 600mm lens; together, they cost $16.000 (€13.700). The difference is obvious, isn't it?
And this is only the basic camera system. What about a tripod, gimbal-head, extra batteries, memory sticks, backpack, filters and other accessories?
So, Is wildlife photography expensive? Well, it depends on how far you want to go: starting from around $3000 and up to more than $16.000, it's all about the budget.