What are the known examples of closed, connected and simply connected manifolds of positive Euler characteristic?

Some points.

1. By Poincare duality the Euler characteristic of a closed odd-dimensional manifold vanishes so we restrict our attention to even-dimensional manifolds. By the classification of surfaces $S^2$ is the only example in dimension $2$.

2. Euler characteristic is multiplicative with respect to products (e.g. by the Kunneth theorem). So we can find examples of manifolds with positive Euler characteristic by taking products of an even number of manifolds of negative Euler characteristic together with any number of manifolds of positive Euler characteristic. The Euler characteristic is also multiplicative with respect to (nice) fiber bundles so we can consider nontrivial fiber bundles with suitable bases and fibers also.

3. The Euler characteristic of a connected sum of closed $n$-manifolds satisfies the "inclusion-exclusion" formula $\chi(M\#N) = \chi(M) + \chi(N) - \chi(S^n)$. This means when $n$ is even, connected sum with $N$ increases the Euler characteristic iff $\chi(N) \ge 3$. Also the connected sum of simply connected manifolds is simply connected.

4. Any manifold whose cohomology is concentrated in even degrees (say, over $\mathbb{Q}$) has positive Euler characteristic, and there's a large source of examples of such manifolds coming from algebraic geometry: every generalized flag variety $G/P$ over $\mathbb{C}$ has this property. $\mathbb{CP}^n$ is a special case of this construction but we also have Grassmannians and complete flag varieties, for example, which are also simply connected (probably generalized flag varieties over $\mathbb{C}$ are always simply connected but I don't know how to prove it).

5. By Poincare duality, any closed simply connected $4$-manifold has cohomology concentrated in even degrees and hence has Euler characteristic at least $2$. This blog post on hypersurfaces in $\mathbb{CP}^3$ discusses their topology; in particular they are completely classified up to homotopy (Milnor, Whitehead) and even up to homeomorphism (Freedman). Their Euler characteristics can be arbitrarily large: the linked post shows that the Euler characteristic of a smooth degree $d$ hypersurface in $\mathbb{CP}^3$ is $d^3 - 4d^2 + 6d$. When $d = 1$ we get $\mathbb{CP}^2$ which has $\chi = 1 - 4 + 6 = 3$, when $d = 2$ we get $\mathbb{CP}^1 \times \mathbb{CP}^1$ which has $\chi = 8 - 16 + 12 = 4$, and when $d = 4$ we get a K3 surface which has $\chi = 64 - 64 + 24 = 24$.

6. For the classification of closed simply connected $6$-manifolds see here. Wall showed that any such manifold splits as a connected sum of a number of copies of $S^3 \times S^3$ ($\chi(S^3 \times S^3) = 0$ so this connected sum lowers the Euler characteristic by $2$ and removing it increases the Euler characteristic by $2$) and a manifold $M$ with $b_3 = 0$, hence $M$ has rational cohomology concentrated in even degrees and so has positive Euler characteristic. Incidentally, this makes a connected sum of two copies of $S^3 \times S^3$ the simplest example of a closed simply connected manifold with negative Euler characteristic, so now we can actually carry out the construction I suggested in point 2: the product of two such sums is a closed simply connected $12$-manifold with positive Euler characteristic whose rational cohomology is not concentrated in even degree.

7. The Hopf conjecture claims in part that a closed even-dimensional manifold admitting a metric with positive sectional curvature has positive Euler characteristic. In dimension $2$ this of course follows from the Gauss-Bonnet theorem, and as Wikipedia discusses this also holds in dimension $4$.

8. If I were looking for more examples I might look through lists of symmetric spaces.

Qiaochu has given a very nice answer, but I wanted to add another large class of examples.

Suppose $G$ is a simply connected closed Lie group, and $H$ is a connected closed Lie group. Suppose $H$ acts on $G$ freely via some action, and suppose further that the rank of $H$ is equal to the rank of $G$. Then the orbit space $G/H$ is a closed, simply connected manifold of positive Euler characteristic.

One way to construct such actions is as follows. Beginning with any $G$ as above, let $H$ be a connected subgroup of $G$ which contains a maximal torus of $G$. Then $H$ acts on $G$ by left multiplication, and this meets all the hypothesis above. This gives rise to the so-called homogeneous spaces.

Another way to construct such actions is to allow $H\subseteq G\times G$. Then $H$ acts on $G$ via $(h_1,h_2)\ast g = h_1 g h_2^{-1}$. When this action is free, this gives rise to the so-called biquotients.

The theorem above doesn't require that $H$ acts on $G$ using the multiplicative structure on $G$, but I don't know of any examples that are not of this type.

The proof of the theorem is as follows. First, the fact that the orbit space under a free action by a compact Lie group is a manifold is well known. So let me focus on showing that the quotient is simply connected and that it has positive Euler characteristic.

Since $H$ acts on $G$ freely, there is a principal $H$-bundle $H\rightarrow G\rightarrow G/H$. The long exact sequence in homotopy groups associated to this ends with $$...\rightarrow\pi_1(H)\rightarrow \pi_1(G)\rightarrow \pi_1(G/H)\rightarrow \pi_0(H)\rightarrow ...$$

By assumption, $\pi_1(G)\cong\pi_0(H)\cong 0$, so it follows that $\pi_1(G/H) = 0$.

Lastly, the fun part. Why does $G/H$ have positive Euler characteristic? Well, all Lie groups have the rational homotopy groups of a product of odd spheres. Specifically, $\pi_{even}(G)\otimes \mathbb{Q} = 0$ and $\dim \pi_{odd}(G)\otimes \mathbb{Q} = \operatorname{rank}(G)$.

From the long exact sequence in rational homotopy groups associated to the bundle $H\rightarrow G\rightarrow G/H$, it follows that $G/H$ is rationally elliptic. Further, from the same exact sequence together it follows that $\dim \pi_{even}(G/H)\otimes \mathbb{Q} = \dim \pi_{odd}(G/H)\otimes\mathbb{Q}$.

For rationally elliptic spaces, this condition on rational homotopy groups forces $\chi(G/H) > 0$. See, for example, Felix, Halperin, and Thomas's book "Rational Homotopy Theory", specifically in Part VI (section 32).