Are there any plans for the 'Hubble2'? Maybe with an even bigger mirror capable of even finer resolution, to look even deeper into the origins of the universe?
If so, when can we expect to see it in action?
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The James Webb Space Telescope (JWST) will be Hubble 2.0, and, my personal favorite, it could get a better look at planets in other solar-systems than we've ever had. Scheduled for launch in 2018. (Footnote, as Rob Jeffries points out, it's infra-red through long-wave visible light only). Hence, no Blue/Indigo/Violet or Green, but some yellow. So, it's not quite like Hubble, but it's still often called Hubble's replacement.
Chandra observes X-rays.
Spitzer observes infra-red. They both followed Hubble, but they're not the same type of telescope.
WFRIST/AFTA is probably the closest thing to Hubble that is going up. It has a similar 2.4 meter mirror will perform wide-field imaging and surveys of the near infrared.
It will have a Wide Field Instrument (WFI) which operates in the 0.7 - 2.0 micron range as well as a Coronagraphic Instrument (CGI) which operates in the 0.4 - 1.0 micron range.
Since most of the telescope hardware (called AFTA (Astrophysics-Focused Telescope Assets)) was donated by the National Reconnaissance Office (NRO) to NASA, a lot of the telescope hardware already exists and does not need to be built so it's likely to be a 'quickly built' mission.
Although the JWST is often touted as the successor to the HST, this isn't technically true, as Rob pointed out the JWST is designed to work in the infrared not optical wavelengths. One possible, more direct, successor could be the ATLAST (Advanced Technology Large-Aperture Space Telescope) telescope, it would not only be able to observe in the optical but also infrared and ultraviolet. The primary mirror is planned to be between 8-16.8m in diameter and would be able to achieve up to 2'000 times better angular resolution than the Hubble can at present. It would make use of the newly developed Space Launch System (SLS) to place it into an L2 orbit (also where JWST is heading), and there are also plans to allow servicing missions to the telescope every 5-7 years (similar to how the HST was serviced in orbit) to enable astronomers to upgrade the instruments on-board.
However this telescope still hasn't left the drawing board and is unlikely to meet its 2025 launch date, funding for such a large telescope is guaranteed to cause a delay in construction as it would probably require an international collaboration just like the HST was jointly commissioned by ESA and NASA.
See:http://www.stsci.edu/atlast for references.