Orthodoxy in Taiwan
Orthodox Christianity has a small presence in Taiwan, with a total following of a hundred or so. The church arrived on the island around 1895, but its history has not been continuous.
Japanese era (1895-1945)
After the Qing Dynasty ceded Taiwan to Japan in 1895, the first Orthodox believers--who were Japanese immigrants--arrived on the island, and almost immediately began petitioning St. Nicholas (Kasatskin), Archbishop of Japan to send them a priest. In 1901, a Tokyo synod created the Christ the Savior Parish in Taiwan. Its first priest was Fr. Simeon (Okava or Yukava--spellings differ), followed (in 1911) by a Fr. Titus (Kariyama). Records indicate a Taiwan-based Orthodox population of 15 or 17 (in 1900), 29 (in 1901), and 44 (in 1903). The activity of the community was interrupted by the 1912 death of St. Nicholas of Japan, and largely ceased with the end of Japanese rule in 1945.
White Russian era (1949-1970's)
In 1949, some 5000 Russians arrived from China (e.g. Shanghai, Harbin, Xinjiang) in the wake of the Chinese Civil War, and began gathering in Taipei's Cafe Astoria. Mention is made of a Korean War-era funeral led by Bishop (later Archbishop) John (Shahovskoy) of San Francisco, then a U.S. army chaplain en route from Korea to the USA. Archbishop Ireney (Bekish) of Tokyo (later New York) made annual visits to Taipei between 1957 and 1959, celebrating divine liturgy in a private home, called the Church of the Forerunner. In 1960 he ceded these duties to an American military chaplain, Fr. Nikolay Kirilyuk. 1965 saw a visit by Metropolitan Vladimir (Sagosky) of Japan (later San Francisco), American military chaplain Archpriest Peter Zurnovich, and Fr. Kirill Arihara. The number of Orthodox faithful in Taiwan has been variously estimated at 50 (in 1960), 100 (in 1958), and 200 (in 1965). The Russian community's most famous member, ROC President Chiang Ching-kuo's Belarussian-born wife Chiang Fang-liang (née Faina Ipat'evna Vakhreva), did not attend services (and may have nominally affiliated with her husband's Methodism).
Sources differ as the whether this predominantly Russian church had any contact with the earlier wave of Japanese-era believers. By the 1970's the church had again dwindled into inactivity, in part because of a canonical rule requiring the closure of any parish which goes more than fifty years without a resident priest.
Global era (2000-present)
In 2000, a Greek hieromonk, Fr. Jonah (Mourtos) of Gregoriou Monastery (Athos), arrived, under the auspices of the recently-created Orthodox Metropolitanate of Hong Kong and Southeast Asia (OMHKSEA, f. 1996, and affiliated with the Ecumenical Patriarch), and with financial backing from the Kosmas Aitolos Missionary Society of Greece. He had previously been posted to missionary churches in Zaire and Calcutta. A small congregation of perhaps 30 people (swelling to more than a hundred at Christmas and Easter) formed as the Holy Trinity Orthodox Church (Taipei), aka the Orthodox Church in Taiwan, which formally registered with the government in 2003. It originally met in hotels and borrowed Catholic church buildings, then in a rented storefront in Taipei's Tianmu district, before moving to a fourth-floor apartment in Xindian. The congregation has included a mixture of Russians and East Europeans, as well as Chinese and Western converts. Liturgy is conducted in English, with parts translated into Chinese, Russian, and/or Greek. A satellite group, led by a lay reader, has been meeting in Taizhong (Taichung).
In 2005, INTERPOL contacted Taiwan authorities in an attempt to apprehend drug smuggler / Christodoulos aide Apostolos Vavylis, notorious for his role in the church scandals which made worldwide news that year.  Vavylis had been traveling on false identity documents obtained through the assistance of church leaders, including Fr. Jonah, who traveled to Greece to testify to his lack of criminal intent (but was ultimately not called to testify). Vavylis indicated that he had traveled to Italy via Thailand with the help of (unnamed) "Taiwanese friends."  
In 2012, Archbishop Mark of Yegorievsk, head of the Russian church's Office for Institutions Abroad, "reactivated" the (1901) Christ the Savior parish, apparently in response to numerous requests from Russians living in Taiwan. The following year, the Church of the Elevation of the Cross, aka the Taiwan Orthodox Church, was formed as a metochion of the Moscow Patriarchate, with Russo-Canadian hieromonk Fr. Kirill (Shkarbul) as its first resident priest. It meets in first-floor apartment in Taipei's Xinyi District, off the Hulin Night Market. Liturgy is conducted in Russian, and translated into Chinese and English.
Bishop Nektarios (Tsilis) of Hong Kong (OMHKSEA) responded by excommunicating Fr. Kirill and one of his parishioners (both of whom had formerly attended the OMHKSEA mission church), on the charge of uncanonical behavior and "ethno-phyletism."  At issue is whether the Moscow Patriarchate has the right to establish parishes outside of Russia, in what OMHKSEA considers to be territory under the jurisdiction of the Ecumenical Patriarch. The Russian church takes the position that it has the right to operate wherever there are Russians, and points to Moscow's historic privileges in China and Japan (both of which have exercised sovereignty over Taiwan in the past). An OMHKSEA press release specifically rejects arguments in favor of "parallel Orthodox jurisdictions" (as in the USA), adding a note on the political background:
- The Orthodox Metropolitanate knows who is protecting him [Fr. Kirill], as well as all the bad things that he and his collaborators are doing in order to gather followers. We do not want to make any disclosures yet, so as not to scandalise the faithful. [...] Finally, to those who speak of the presence of the Church of Russia in South East Asia and its supposed canonical rights, the Orthodox Metropolitanate of Hong Kong and South East Asia states once again its unwavering and clear position that the presence of the Church of Russia in South East Asia is uncanonical and that any decision of the Synod of the Church of Russia concerning the Far East is considered invalid. At some future point, the Orthodox Metropolitanate will comment on the so-called historical arguments presented by the Church of Russia to support its uncanonical actions. 
(See also Fr. Jonah's comments likening certain ethno-phyletists to Judas and the KGB.)
In the Taiping District of Taichung, there is said to be a house church belonging to the Orthodox Church in America (OCA). If so, it would constitute a third rival Orthodox jurisdiction on the island.
Note on Taiwan's Religious Environment
Most of Taiwan's population follows the Chinese folk religion (which is known by various names, including both "Buddhism" and "Daoism"). Unlike Western religions, participation in the Chinese folk religion does not usually involve required beliefs or behaviors, or a formal group identity, and is difficult to separate from such borderline religious phenomena as fortune-telling, qigong practice, holiday observances, or ancestor veneration. A minority (perhaps 5 or 10 percent) are "Buddhists" in the more international sense of having taken refuge with a Buddhist monk or nun. Christians apparently comprise 4 or 5 percent of the population, roughly half of them Protestant (with Presbyterianism by far the largest denomination) and half Catholic.
Dutch Calvinist and Spanish Dominican missionaries arrived on "Formosa" (as Taiwan was then called) within a few years of each other in the 17th century, but failed to establish permanent congregations before the expulsion of the Spanish (by the Dutch) and Dutch (by the Ming loyalist / pirate lord Koxinga). During the 1860's, Dominicans (from the Philippines) and Presbyterians (from England and Canada) established relatively successful missions, with immense consequences to the island's culture and development. For example, one Canadian Presbyterian missionary, George Leslie Mackay, founded Taiwan's first university and hospital, and created a script for the Taiwanese Minnan language.
New foreign missions were forbidden entry during the Japanese era, indirectly benefitting the Catholics and Presbyterians, who were already present on the island. After 1949, a wave of missionaries from various denominations (including the Baptists, Episcopalians, Lutherans, and Adventists, among many others) arrived in Taiwan from China, and established churches representing those denominations. The Presbyterians enjoyed further growth during the 1950's and 1960's, when the church took an outspoken stance in favor of human rights and Taiwan independence. The end of martial law in 1987 has also liberalized the religious environment, with Pentecostalism and the House Church movement being important new trends.
Simeon Eryshev (Семён Epышeв), Православнaя Mиссиe иa Тайва́ньe--Историe, coвpeмeниocтъ Перспективы (Orthodox Missions in Taiwan: Historical and Contemporary Perspectives). MDiv thesis, Moscow Theological Academy and Seminary, 2005.
- http://www.orthodox.cn/contemporary/sayama_en.htm http://www.orthodox.cn/news/20080826nikolaisayama_en.htm
- http://www.orthodox.cn/localchurch/taiwan/1958-1959ireney_en.htm http://orthodox.cn/localchurch/taiwan/glebrar_en.htm http://www.orthodox.cn/localchurch/taiwan/19650120vladimir_en.htm
- Prof. Andr. Demetropoulos, Personal Life of Clerics. National Kapodistrian University of Athens, School of Law, 2004-20055. http://www.greeklaws.com/pubs/uploads/1216.pdf
- http://www.omhksea.org/2013/06/excommunication-of-the-schismatics-in-taiwan/ http://www.omhksea.org/2013/07/press-release-about-the-schismatics-in-taiwan/
Cf. the parallel Wikipedia article, "Orthodox Christianity in Taiwan."