I can sooooo empathize with you! My dd (23 months) is also an insane climber. When we were given a dresser, I came in my dd's room to find her sitting in the middle drawer watching Sesame St. After a pic, I redirected her while doing my best to hide my laughter. She also cannot get into a chair the normal way...she has to climb over the arm of it. Thankfully, my lil one seems to have reasonably good balance.
Here is what we have done with my crazy little dd:
We bought her a trampoline...small one from Walmart $30. She loves it.
I try to take her to a park where she can climb on the equipment as often as possible. Of course, I am always within arms reach.
I have rearranged my house several times to make it safer for her to explore. She has already learned how to climb over the couch into the office area, though. Sigh.
I have deliberately made her room safe...or as safe as I possibly can considering the ingenuity of my dd. She will take toys into her room to use to climb on to get to higher reaching objects.
Baby gates work well, but I would invest in several. If it looks like they will be able to climb over those, I would put two in one doorway. We haven't gotten to this yet, thank goodness.
Check out some thrift stores in your area. I have found several larger toys/activity centers there for a fraction of the price of retail. You may find something there that will help divert her.
I have found one thing that my dd will stop ANYTHING for...she loves the Sesame Street first words flash cards from the Dollar Tree. She will stop whatever she is doing and come directly to me, climb on my lap and we will go through them for a while.
One realization that I have come to, though, is that my dd will deliberately do things she KNOWS she isn't supposed to do...climb into the office area, tamper with the thermostat, try to get to the tv, etc...when she is tired for her nap or it is time for bed at night. I try not to yell or punish her at those times, but directly take her and start her bedtime routine. I also tell her that since it is late and she is doing things she knows are wrong is why she is going directly to bed. I am hoping when the connection hits her of misbehaving leading to bed that she will stop, wanting to stay up slightly later. :-)
Just try to think outside the box...if you come up with ANY other ideas, I would love to hear them. I am always open to new alternatives for dealing with my Houdini.
Good luck.