As noted, GnRH may play a role in adult neurogenesis. Also in the post I linked to:

”[1] Furthermore: “Multiple lines of evidence indicate that the expression of extrapituitary GnRH receptor is not limited to reproductive tissues. For instance, it has been demonstrated by RT-PCR and Southern blot hybridization that the receptor is also expressed in the liver, heart, skeletal muscle, kidney, and peripheral blood mononuclear cells.”[2]

It's a similar story with LH. Wikipedia says:

Luteinizing hormone receptors are located in areas of the brain associated with cognitive function. The role of LH role in the central nervous system (CNS) may be of relevance to understanding and treating post-menopausal cognitive decline.

Some research has observed an inverse relationship between circulating LH and CNS LH levels. After ovariectomy (a procedure used to mimic menopause) in female mice, circulating LH levels surge while CNS levels of LH fall. Treatments that lower circulating LH restore LH levels in the CNS.

This suggests men may want to avoid high levels of hCG for long periods. Also:

LHCGR [luteinizing hormone/choriogonadotropin receptors] have been found in many types of extragonadal tissues, and the physiologic role of some has remained largely unexplored. Thus receptors have been found in the uterus, sperm, seminal vesicles, prostate, skin, breast, adrenals, thyroid, neural retina, neuroendocrine cells, and (rat) brain.

There are still things to be learned about these hormones, which is why I have misgivings about suppressing them indefinitely.