I'm trying to see how far can our star reaches with its gravity. I'm asking if anyone could give info as to what's our star's limit or the furthest object found in our solar system.
The Sun's gravity extends infinitely, but eventually solar objects would be unstable due to the influence of other stars. The minor planet "Sedna" has an orbit which takes it nearly 1000 AU (0.016 light years) from the sun at its furthest point (but now it is a lot closer)
It is also thought that billions of comets must orbit in the outer part of the solar system, out to 50000AU, or 0.8 light years, (or possibly further) forming the Oort Cloud. However, at such distances, they could not be directly observed. This marks the greatest distance at which orbiting solar system bodies can be found.
There's no straight forward answer. In the solar-system, which is well ordered, objects that are in stable orbits, and not too elliptical, have well defined spheres of influence. Planet 9, if/when it's discovered, will probably have the largest sphere of influence for known solar-system objects. Currently, Neptune has the largest.
If the stars near the sun were static relative to each other, the Sun's sphere of influence could be calculated and it would probably extend between 2 and 3 light years. But because the stars are not static, the sphere of influence is constantly changing and stars (probably) exchange outer, loosely orbiting debris fairly frequently.
The Oort cloud by this article is thought to extend to almost 2 light years, so that's one possible answer to your question. If you want to know the most distant aphelion of an object currently orbiting the sun, James Ks answer is good, but I think the outer most aphelion is a bit further than the 0.8 light years that he suggests. At least 2 light-years, possibly even 3. The problem is, an orbit that distant, such an object has a good chance of being deflected before it reaches it's perihelion, a journey that takes over 10 million years. Orbits that distant are likely not very stable. A lot depends on how close other stars get to our sun. A star that passes too close would likely throw everything in the vicinity that it passes through out of wack.
The tiny Scholz's star is thought to have passed within 1 light year of our sun about 70,000 years ago. Stars passing that close are quite rare, but, from the link above
A star is expected to pass through the Oort Cloud every 100,000 years or so. An approach as close or closer than 52,000 AU is expected to occur about every 9 million years.
This does make defining an outermost orbit somewhat difficult, as the most distant orbits take millions of years to reach their closest point from their most distant point, and they run a pretty good chance of being perturbed within a single orbit. Stars likely play Frisbee with their outer-most orbiting objects all the time. Picking an outermost stable orbit is impossible.
A curious sidebar on Scholz, is that, it may have sent a bunch of outer comets and oort cloud objects heading towards the inner solar system. We wont find out how many for another 2 million years or so. That's how long it will take any objects that were sent towards the inner solar-system to reach it.