Himalayas are currently rising about 5 mm/year. What will be the highest point they can reach?
The Himalayas are indeed rising, but they are also being eroded at a comparable rate. It will come as no surprise to anyone that the maximum possible height of a mountain on Earth is only marginally higher than Mt Everest, that is probably about 9000 to 10,000 metres high. There are, of course compressive / convergent plate boundaries where there is a potential tendencey to build higher mountains, but non-linear counter processes kick in to counteract the mountain building. There are three processes, glacial erosion, shear collapse and lower crust / upper mantle plasticity. The latter is the most important. As lateral compression builds higher mountains, the mountain roots are depressed into the hot plastic region such that the mountain literally sinks under its own weight, deforming the root zone, sometimes in spectacular fashion. Go to Scotland, or Norway, where you can see the eroded remains of the long-gone Caledonides, and you can see the exposed and metamorphosed root zone of mountains that were once as big as the Himalayas.
In principle, the mountain would eventually collapse by shear failure, simply because the rock isn't strong enough to support the weight of gigantic mountains. In practice, deformation takes place before the shear failure is reached.
Then there are glaciers. The higher the mountain gets, the more rain turns to snow, then to ice, which is a phenomenally effective erosive agent (just look at any of the worlds recently glaciated terrains - European Alps, Patagonian Alps, Norway, Alaska, New Zealand, etc., to see the spectacular erosion). This erosion peaks at a high altitude, but not at the very highest altitude, where the atmosphere is so cold that it cannot deliver much moisture - so very little snow. Result: glacial erosion takes bites out of the mountain at middle-upper elevations, thereby keeping the slopes steep, whence gravity collapse along pressure release joints, and other related processes. Hence occasional peaks get close to 9000 metres but, in geological terms, they don't last long before erosion brings them down.