I think this question depends on the precise definition of "initial" metrics.
In the bounded curvature case, thanks to Shi's estimate, an initial metric can be well-defined by smooth convergence, i.e., a metric $g(0)$ is said to be an initial metric of the Ricci flow $g(t),t>0,$ if $g(t)\to g(0)$ in $C_{loc}^\infty$-topology.
However, when we consider the nonexistence problem and manifolds with unbounded curvature, the definition of an initial metric is not uniquely defined.
There are at least two cases one can conceive:
- $g(0)$ is complete, $Rm(0)$ is pointwise bounded but not uniformly bounded. In this case, we have more general existence result than Shi, proven by Guoyi Xu in http://arxiv.org/pdf/0907.5604v3.pdf. The convergence is still smooth here.
- $g(0)$ is incomplete, say $Rm(0)$ is pointwise bounded except at one point. Felix Schulze and Miles Simon constructed expanding solitons coming out from certain "cones". http://arxiv.org/pdf/1008.1408.pdf. Their sense of initial metric is of Gromov-Hausdorff convergence, that is, they showed the Gromov-Hausdorff distance of $g(t)$ and $g(0)$ goes to zero as $t\to 0$. For general expanding soliton $g(t)$, the optimal convergence on regular portions is $C_{loc}^{1,\alpha}$(instead of $C_{loc}^\infty$) even both the curvature and volume behave very well. (This can be derived by using harmonic coordinates).
So, to prove a manifold $(M,g)$ admits no short-time solution, we need to indicate which convergence is involved. For example, $(M,g)$ might not be a $C_{loc}^{1,\alpha}$ initial data but indeed a $C^{0}$ initial data for some solutions.
Let's go back to the original question: how to find a manifold (with unbounded curvature) such that NO Ricci flow can converge to it in ANY sense?
It is obvious that we should replace "ANY" by a spectacular term like "Gromov-Hausdorff" before we go further. Even in this case, it is not easy to check, for example, Terry Tao's intuitive example is the one we want. (This is probably due to my poor math ability...)
Actually, Thomas Richard and I constructed an example related to this goal. We found a complete manifold (a rotationally symmetric infinite cusp) which is the last time slice of a Ricci flow $g(t)$. That is, we constructed a Ricci flow $g(t)$ which exists only up to time $T$ and $g(T)$ is a complete smooth metric. In this sense we can say that no solution could flow out from our $(M,g(T))$ as a continuation. But we still don't know can there be a solution, although not a continuation of $g(t)_{t\leq T}$, defining on $t>T$ and converging to $g(T)$ in certain sense when $t\to T$. Our example is similar to (even easier than) Terry's suggestions, however, I haven't conceive a way to show it is "unflowable". In fact, I can imagine a solution flows out from such cusp, hence I guess this question might be harder than people expected.