For this article I go for characters who were either in the party since the beginning of the game, or who clearly had some kind of relation prior to the game's events, but not romantically.

It's often a cliché that the first female in the party is always the love interest of the male hero. But let's be frank... Not all members of the opposite sex we meet do romantically interest us (assuming you are heterosexual). Sometimes we even know each other "too well" for something like that, and would consider each other siblings, or sometimes there is just not that "chemical connection" that makes us fall in love.

Let's list out some of these, shall we?

Etna - Disgaea

In the very first Disgaea game, Etna is the female sidekick to the male protagonist Laharl. Etna is the vassal of Larhal, and as such she has to obey all his order. She appears to hate him with passion, and loves to do everything to mock him, or to even seriously get him hurt. Now when the going gets tough she appears to care more for him than she wants to admit, but it is absolutely clear there is no romantic interest. She even teases Laharl about him doing a pretty poor job hiding his affections for the angel-trainee Flonne.

Sue - Grandia

Sue is Justin's childhood friend. See my: female childhood friends article for full details.

Lucca - Chrono Trigger

Lucca is in the same league as Sue, but I had to mention her here.

Grubb - Septerra Core

A man in this list? Well Septerra Core has a female protagonist, so a man had to fill this list in the case of that game. The game begins with Maya alone, however it's clear from the start that Grubb and Maya have known each other for a long time, also evidenced in how you see the harmony between the two in the opening video, and also in how Maya and Grubb communicate when he is the first character to join the party (together with his robotic dog: Runner). It's clear Grubb cares for Maya, also evidenced when he comes to her defense when she's sexually harassed by NPC characters, but it's also clear there is no romantic interest from either side. When the girl Led pops up, later in the game game, she openly dares to speak out to Maya that she actually likes Grubb. Maya, who could see right through Grubb's insults towards Led, already knew Grubb returns Led's affections, and even gives her the advice to have some patience on him, making it clear Maya is not jealous or anything, but in the same time she knows him very well. So we can safely come to the conclusion that Maya and Grubb are close to each other, but not in a romantic perspective.

Miranda - Grandia III

I was a bit conflicted about adding her, but truth is, she is after main character Yuki the first character the join the party, plus she's known him from since he was very young, and no there is no romantic interest, and the game would be very controversial if there was. Miranda is after all, Yuki's mother. Although Yuki never refers to her by words as "mother" or "mom", but always by her first name, as well when her addresses her in a conversation as when he mentions her to other people. Although Miranda is quite a tomboy and a woman with nerve and a very capable fighter, she does show herself in the typical characteristics of a mother, such as scolding Yuki for any mischief, and trying to get him a girl-friend by manipulating him and Alfina close to each other.
Say what you want, she is of the opposite sex (from Yuki's point of view) and she is not romantically interested, so she fits the description of this list.