An example is the "complete rainbow graph without monochromatic triangles". Let $L = \{R_i : i \in \omega\}$ be a language consisting of $\omega$-many binary relation symbols $R_i$, and take the class $\mathcal{K}$ of finite $L$-structures $C$ such that

 - each $R_i^C$ is a graph relation (that is, irreflexive and symmetric);
 - the $R_i^C$s are disjoint;
 - every ordered pair of distinct elements of $C$ lies in some $R_i^C$;
 - no $R_i^C$ contains a triangle.

We think of each $R_i^C$ as consisting of the edges of $C$ of colour $i$.

Then $\mathcal{K}$ is a strong amalgamation class (join the ears of the finite amalgam with edges of a colour that's not used in either ear). Let $M$ denote its Fraïssé limit, and let $\mathcal{U}$ denote the class of $L$-structures which embed into $M$ (so including countably infinite ones).

We will see that $\mathcal{U}$ does not have amalgamation.

Let $A \in \mathcal{U}$ be countably infinite. Then $A$ has two adjacent edges $a a_0, a a_1$ of different colours, as there are no monochromatic triangles in $A$ and $A$ is a complete graph. Enumerate $A$ as $a, a_0, a_1, a_2, \cdots$. Let $i_1$ be the colour of $a a_0$ and let $i_0$ be the colour of $a a_1$ (this is not a typo!). Enumerate the set $\omega$ of colours as $i_0, i_1, i_2, \cdots$.

Define structures $B, C$ as follows. Let $B$ consist of $A$ together with a new point $b$ such that $ba$ has colour $i_0$ and $ba_j$ has colour $i_j$ for $j \geq 0$. Let $C$ consist of $A$ together with a new point $c$ such that $ca$ has colour $i_1$ and $ca_j$ has colour $i_j$ for $j \geq 0$. Then $B$ and $C$ are complete and contain no monochromatic triangles, so lie in $\mathcal{U}$.

Then $B, C$ cannot amalgamate over $A$, as $b$ and $c$ cannot be identified in the amalgam due to their different coloured edges to $a$, and any coloured edge between $b$ and $c$ would form a monochromatic triangle.