(CNN) — House Republicans' failure to elect a speaker on Tuesday after multiple rounds of voting isn't just denying the GOP a leader — it's holding up much of the functioning of the chamber.

The position is traditionally filled on the first day of a new Congress, followed by the swearing in of new members, but with the floor fight spilling into Wednesday, members-elect have yet to take the oath of office.

Incoming lawmakers arrived on the floor on Tuesday with their families in tow, expecting to pose for a photo and get started with their first day as lawmakers, but were instead greeted with a several-hour-wait as the speaker election went to multiple rounds of balloting — the first time that's happened in 100 years.

Every new Congress must pass a new set of House rules, so without a speaker to oversee adoption of those rules, none will technically exist.

Without an approved House Rules package by the end of business on January 13, committees won't be able to pay staff, according to a letter sent last week by the committee in charge of administrative matters, which was first reported by Politico and obtained by CNN.

The same memo warned that student loan payments for committee staff wouldn't be disbursed if a rules package isn't adopted by mid-January.

It's just one of the many ways a battle over the next speaker could paralyze the House and the Republican majority from operating efficiently in their opening days with some of the harshest penalties falling on rank-and-file staffers.

For committees whose chairs aren't known, they will be headed up in the interim by the committee's senior-most Republican who also served on the panel in the last Congress, according to the letter sent last week.

But without fully functioning committees, to amend and approve bills before they make their way to the floor for a vote, there will be little legislating. That means Republicans may also have to wait before tackling some of their most pressing priorities, including investigations into President Joe Biden's administration and family.

Outside of the speaker's role effectively running the House, they are also in the line of succession for president — raising questions about what happens if there's no one in the position that's second in line for the presidency after the vice president.

The Senate president pro tempore is third in line. Sen. Patty Murray was elected to that role Tuesday, making the Democrat from Washington the first woman to hold the position.

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