Does a lens with this zoom range exist?

No.

Putting aside the engineering issues with designing a zoom for such a large range, I'd like to put forth a frame challenge: What do you think you want a 2600mm lens for?

DSLR lenses of longer than 400mm are generally regarded as "exotic" because of their limited utility. Lenses up to 800mm or so are attractive for wildlife photography, airshows, and certain sports, but even in those fields the workhorse lenses fall in the more common 100-400mm range. Lenses beyond 1000mm are exceedingly rare for any purpose.

There are technical reasons for this, but I think they're largely a distraction. More important are some practical and artistic concerns:

1. Atmospheric distortion and haze: It is difficult to obtain a high-quality image of something that is far away regardless of lens quality simply because of the atmosphere between the lens and the subject.
2. Perspective and composition: Subjects that are far away have a perspective that makes clear to the viewer that they are far away - telephoto lenses make things look bigger, not closer. This tends to result in images that are unengaging, giving rise to the popular (though linguistically shaky) advice to "zoom with your feet".

Real DSLR "superzoom" lenses have zoom ratios around 15x to 18x, with maximum focal lengths of around 300-400mm. They have some utility for travel photography, but in my experience they are often an indication that the user has purchased a camera with the wrong tradeoffs for their use case.

There's a reason no one makes even a 24-200mm FF lens. Several, in fact.

• The main one is that not many photographers who know what they are doing would ever consider buying such a lens for a FF camera, particularly for one with as high resolution as most FF cameras offer in 2021.
• To get anywhere approaching a constant, usable aperture the lens would be very heavy and large.
• Such a lens would be very expensive to produce at any level of decent image quality.
• Lenses with smaller zoom ratios can be smaller, lighter, cheaper, faster, and produce higher image quality than a larger, heavier, more expensive, slower lens with inferior image quality.

The entire point of an interchangeable lens system camera is to allow you to use different lenses that are better or even great at one thing but unsuitable for other things. Fixed lens cameras force you to use a single lens that is mediocre or worse at a lot of things but better at nothing. Insisting on using a single lens for everything on an interchangeable lens camera is not much different than using a fixed lens camera. In some cases the fixed lens camera may meet your needs better than an ILC with only one lens.

The best lenses are all prime lenses. That means a single focal length. No.Zoom.At.All. They're really good when they provide the field of view and other characteristics you need. This is because they can be optimized to do one thing at one focal length. A good flat field 100mm macro lens is different from a good 85mm, 105mm, or 135mm portrait lens. But lenses optimized for doing one thing very well are usually not very flexible, so you need a lot of them for various different things. Some are pretty good for not much money (e.g. EF 50mm f/1.8 STM @ $120). Others are incredibly good for a boatload of cash (e.g. EF 400mm f/2.8 L IS II @ $10K). Most fall somewhere in between.

Compared to their zoom lens counterparts, in addition to equal or better optical quality at a lower price prime lenses can also be smaller/lighter, have wider maximum apertures, and often still be much cheaper than zoom lenses in the same focal length range.

Short ratio zoom lenses, that is zoom lenses with a less than 3X difference between their longest and shortest focal length, can also be very good. But the best ones cost a lot.

When you move outside of the 3x limit is when image quality really starts to noticeably go down. Some 4-5X zoom lenses that fall entirely in the telephoto range can be pretty good. But when you start trying to design a lens that goes from wide angle to telephoto and covers a 5X-10X or more zoom range, that is when it really starts getting difficult to keep it affordable and manageable with regard to size and weight and still provide excellent image quality. You'll usually get better image quality and spend less buying something like an 18-55mm and a 55-250mm pair of zoom lenses than you would get with an 18-200mm 'all-in-one'.

I want to buy a zoom lens for a DSLR but I want its Focal length to start from a small number (18, 55, 100, 200) and end to a very large number (1300, 2600) so it can see near and far. Also I want it to be small to carry it with me.

No such lens is currently available for any DSLR.

The reason there are no such lenses are that they would be far too heavy, much too large, and way too expensive to be practical while still delivering much poorer optical image quality than much smaller, lighter, and cheaper lenses can deliver.

If you really want to go there, you can get a broadcast quality video lens such as the Canon DigiSuper 100AF that projects an image circle large enough for a 2/3" broadcast camera with a 9.3-930mm focal length. It gives an equivalent field of view on such a cameras as that of a 36-3656mm lens on a FF camera. It only weighs 60 pounds and costs a bit more than $200,000. And that is for a 9.59x5.39mm video sensor with a 3.9X crop factor.

For a Full Frame sensor it would need the front element to be 4X as wide, 4X as tall, and be 4X as long. It'd probably weigh about 64X as much (Each lens element would have 64X the volume when 4X larger in the three linear dimensions), and maybe 4,096X as much (i.e. $819 Million USD). If you're asking for the Defense Department of a major industrial power, they might be willing to make one for you?

I think the closest contenders are the tamron 18-400 aps-c lens or the sigma 60-600 full frame lens. None of these cover nerly the zoom range you are asking for but are aproaching the use cases you are talking about. Slap on q 2x teleconverter on the sigma 60-600 and you are at 120-1200 focal legth with full frame coverage. I would not expect good autofocus or stelar sharpness with this combo thou. A better idea might be the 60-600 with an 1.4 teleconverter and a good aps-c body such as canon 7d mark ii

I'm kind of repeating what's already been said, but I just want to put another spin on it.

It sounds like you want the superzoom capabilities of a compact camera, but "ramped up" to suit your DSLR – presumably to maintain some of the advantages of using a larger sensor. Unfortunately you can't translate the superzoom of a compact camera to your DSLR without the lens also incurring a corresponding increase in size. This makes it heavier and more expensive (and, in this case, non-existent). You want the lens to be small to carry it with you? Well, sorry, but the laws of physics disallow it.

As Michael mentioned, exchangeable-lens cameras exist for the exact reason that you can change lenses. If you don't like to change lenses, and don't like missing photo opportunities because you've left your large, heavy camera system behind, then maybe you just have the wrong camera for your needs. The Nikon P1000 has been mentioned in the comments – I'm not familiar with this model – the one I was going to suggest is the Sony RX10 IV with 25x optical zoom. You simply can't get some fantastical tiny lens with huge zoom and amazing image quality.