I am trying to find some examples of Fraïssé classes that do **not** have the strong amalgamation property. Anyone?



A Fraïssé class without the strong amalgamation property.
**In this edit, the relevant theorem is properly stated, and a previous "fake" example has been replaced by a genuine one (in light of corrective comments by Joel Hamkins and Emil Jeřábek).**


Goldstern and Hamkins have provided nice examples. I will add:


**Theorem** *A Fraïssé class has the strong amalgamation property if
and only if the automorphism group of its Fraïssé limit $M$ has no algebraicity (i.e, the algebraic closure of any finite set X within $M$ coincides with X itself).*


The above theorem appears as Exercise 8 of sec. 7.1 of Hodges' text on Model Theory, as well as [(2.15), p. 37] of Peter Camerons's 1990-monograph Oligomorphic permutation groups. According to the Hodges text, it is due to James Schmerl (Journal of Symbolic Logic, 45, pp.585-611, 1980). 



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> So a "natural" example of a Fraïssé class without the strong amalgamation property is obtained by fixing a prime number $p$, and considering the class of finite fields of characteristic $p$. It is well-known that the Fraïssé limit of this class is the algebraic closure of the prime field of characteristic $p$.
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