Edited to fix the example, as per Zack's suggestion.
Edit 2: So it turns out that when I think 'manifold' I tend to assume the nicest possible object. As I believe is standard, I would like to assume that all manifolds are 2nd countable and Hausdorff. Furthermore, let's say that our manifolds are connected and closed.
The Whitney embedding theorem states that any smooth $n$-manifold may be smoothly embedded into $\mathbb{R}^{2n}$. If we consider embeddings into more general $k$-dimensional manifolds, is it possible find a '$n$-universal' manifold of dimension less than $2n$?
For example, a non-orientable 2-manifold cannot be embedded into $\mathbb{R}^3$, demonstrating the sharpness of the Whitney embedding theorem.
However, there are 3-manifolds into which we can embed any surface, such as $M = \mathbb{RP}^3 \sharp \mathbb{RP}^3$. Indeed, by the classification of surfaces we know that any surface may be decomposed as a connected sum of copies of $\mathbb{RP}^2$ and tori. In fact, by the monoid structure of closed surfaces under connected sums we may take this sum to have at most 2 copies of the projective plane. Now, embed 2 disjoint copies of the projective plane into $M$ and arbitrarily many copies of the torus. Taking the connected sum of these we see that any closed surface is embeddable into $M$.
Can we do something similar in higher dimensions?
Best Answer
Yes, the Whitney theorem can be improved in many cases. For example, C.T.C. Wall proved all 3-manifolds embed in $\mathbb R^5$.
Precisely what is the optimal minimal-dimensional Euclidean space that all $n$-manifolds embed in, I don't know what the answer to that is but Whitney's (strong) embedding theorem is only best-possible for countably-many $n$, not for all $n$. See Haefliger's work on embeddings -- I believe he noticed many cases where you can improve on Whitney.
The suggestion to improve Whitney's theorem that you're giving -- making the target not a Euclidean space but a manifold -- in a sense you're asking for something much weaker than Whitney's theorem. For example, given any $n$-manifold, you can take its Cartesian product with $S^1$. Take the connect sum of all manifolds obtained this way. It's a giant, non-compact $(n+1)$-manifold, and all $n$-manifolds embed in it. This isn't so interesting.