TSMC and Samsung are also exploring EUV, so it's not just Intel. New lithography is necessary to continue shrinking, otherwise mask costs will rise exponentially (literally).
To figure this out, draw a square, and each node is an exposure.
This process works forever in theory, but is limited in practice by the mutual registration of the exposures.
See here: https://en.m.wikipedia.org/wiki/Multiple_patterning
Edit: the exponential rise happens because to go 2x smaller I now need to double my # of masks. Of course, in practice overlay accuracy could kill you first.