As Fianna Fáil and Fine Gael are considered likely to go into government together, the question is whether they will form a coalition with smaller parties or a block of Independents.
Amongst those newly re-elected TDs is Seán Canney who was poll topper in Galway East.
He believes there are up to 10 Independents open to supporting the two main parties.
The number needed to form a majority in the next Dail is 88.
Whether it's Independents or others that get the two main parties comfortably over the line, it will take some time and a lot of talking.
Catherine Connolly was a poll topper in Galway West and says her preference is for a "left government" but that she would take whatever role she could and that she would "embrace power".
However she added: "I don't think Fianna Fáil and Fine Gael are going to seriously engage with me in relation to a radical change in housing policy, in relation to using our neutrality as a powerful voice for peace in the world, stop the troops in Shannon. I don't think that's going to happen. Should they change their minds I'd be absolutely delighted to hear from them."
Ms Connolly said the lack of trust now was a "huge threat to democracy" and that she would continue to articulate what people are saying to her on the ground.
She says the performance of herself and Sinn Féin's Mairéad Farrell shows the narrative has changed hugely.
"Two 'radical women' have topped the poll which shows they are crying out for change in Galway and that has been my experience for three elections".
She said she wanted to talk about issues rather than numbers and cited the housing crisis, the traffic congestion "choking our city" and what she called the existential threat of climate change.
Deputy Connolly quoted the Housing Commission's call for a "radical reset" of policy and described as "obscene" that 15,000 people are homeless. "That's viewed as collateral damage to a successful housing policy."
She said the role of the opposition is "greatly minimised" and that she would happily continue to be a "powerful voice" to that end.
Independent TD Marian Harkin won the fourth seat for Sligo-Leitrim after a late surge of transfers. She has also said she was open to supporting a Fianna Fáil/Fine Gael coalition.
"Fianna Fail and Fine Gael are 'within shouting distance' and I will certainly be one of those people speaking to them and talking and negotiating. That's the reason I ran for election in the first place."
She said she has spoken to a number of people but has promised not to divulge individual telephone calls because everything is "speculation" at this point.
A process in coalition negotiations - Canney
Mr Canney said there is a process in any coalition negotiations and Fianna Fáil and Fine Gael have to set out what they are going to do to try to formulate a stable government.
He said "my track record. I was in government in 2016 to 2020 as an Independent. We were very stable at that time..."
He added that "the last time I did this, there was a block of TDs. There was Denis Naughten, Katherine Zappone and there was Shane Ross, Finian McGrath, John Halligan, Kevin Boxer Moran and I.
"And we went into government at that time on the basis that the country needed a government. And I think at this stage I am prepared to do that again if need be.
"But it will all be contingent on what the Programme of Government would look like."
Mr Canney said the independents’ demands will depend on their constituencies but will have wider issues of concern too.
Separately, Kevin 'Boxer' Moran, Independent TD for Longford-Westmeath, said there are plenty of small parties that can go into government and that "we all want to govern and that is why we put our names forward".
'This idea of saying you don't want to in case you get burnt. One day in government is better than 100 days in opposition. I've been there."
Independent Ireland TD Michael Fitzmaurice, for Roscommon-Galway, said he has not heard from any of the three main parties.
He said a lot of "shadow boxing" is going on and he believes Fine Gael will play hard ball for a while in order to get "fifty-fifty on everything".
He said that the focus must be on forming a stable government.
He added that Independent Ireland is open to sitting down and holding talks with parties but that they would have to focus on policy.
