They had one. The US overthrew it and installed a puppet, the Shah, during the 1950s.
Their actions have returned to bite them in the backside.

@Petter Before the Shah they had a Russian backed socialist puppet. They started to have a good relationship with the Russians. It was an intolerable situation!

Whilst on the subject of non religious governments, Iraq had one until Bush and Blair dreamed up invisible weapons of mass destruction and overthrew it.

Also be nice were there less religion in the US government, of course.

@zesty Not correct. Mosaddegh, the secular prime minister, was openly disdainful of Communism and Russia.
I ask you to read the following summary (Please, please do!) and whilst doing so also to keep in mind that modern day Iraq and Iran are in the same "oil rich area".

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Many Iranians regard Mosaddegh as the leading champion of secular democracy and resistance to foreign domination in Iran's modern history.
An author, administrator, lawyer and prominent parliamentarian, his administration introduced a range of social and political measures such as social security, land reforms and higher taxes including the introduction of taxation of the rent on land. His government's most significant policy, however, was the nationalization of the Iranian oil industry, which had been built by the British on Persian lands since 1913 through the Anglo-Persian Oil Company. (Later British Petroleum and BP)
Initially, the USA opposed British policies. After mediation had failed several times to bring about a settlement, American Secretary of State Dean Acheson concluded that the British were "destructive, and determined on a rule-or-ruin policy in Iran."
The American position shifted in late 1952 when Dwight D. Eisenhower was elected U.S. President.
In November and December, British intelligence officials suggested to American intelligence that the Persian prime minister should be ousted. British prime minister Winston Churchill suggested to the incoming Eisenhower administration that Mossadegh, despite his open disgust with socialism, was, or would become, dependent on the pro-Soviet Tudeh Party, resulting in Iran "increasingly turning towards communism" and towards the Soviet sphere at a time of high Cold War fears.

(Remind you of anything? Iraq and "Weapons of mass destruction", perhaps?)

In early 1953, the United States and the United Kingdom agreed to work together toward Mosaddegh's removal and began to publicly denounce Mosaddegh's policies for Iran as harmful to the country
According to Dr. Donald N. Wilber, who was involved in the plot to remove Mossadegh from power, in early August, Iranian CIA operatives pretending to be socialists and nationalists threatened Muslim leaders with "savage punishment if they opposed Mossadegh," thereby giving the impression that Mossadegh was cracking down on dissent earlier than planned, and stirring anti-Mossadegh sentiments within the religious community.
In 2013, the U.S. government formally acknowledged the U.S. role in the coup, as a part of its foreign policy initiatives.

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The above is a shortening of this entry.
[en.m.wikipedia.org]

@zesty After this engineered coup, the Shah became a puppet monarch, and the land became prey to religious leaders, eventually leading to their total domination of both the country's government and its dogma.