We consider the mass critical two dimensional nonlinear Schrödinger equation (NLS) i∂tu + ∆u + |u| 2 u = 0, t ∈ R, x ∈ R 2. Let Q denote the positive ground state solitary wave satisfying ∆Q − Q + Q 3 = 0. We construct a new class of multi–solitary wave solutions: given any integer K ≥ 2, there exists a global (for t > 0) solution u(t) of (NLS) that decomposes asymptotically into a sum of solitary waves centered at the vertices of a K–sided regular polygon and concentrating at a logarithmic rate as t → +∞ so that the solution blows up in infinite time with the rate ∇u(t) L 2 ∼ | log t| as t → +∞. This special behavior is due to strong interactions between the waves, in contrast with previous works on multi–solitary waves of (NLS) where interactions do not affect the blow up rate. Using the pseudo–conformal symmetry of the (NLS) flow, this yields the first example of solution v(t) of (NLS) blowing up in finite time with a rate strictly above the pseudo–conformal one, namely, ∇v(t) L 2 ∼ log |t| t as t ↑ 0. Such solution concentrates K bubbles at a point x0 ∈ R 2 , i.e. |v(t)| 2 ⇀ KQ 2 L 2 δx 0 as t ↑ 0.