Brennan was born in Dublin, his family moved to Switzerland when he was seven years old. He took piano lessons from age eleven, but played bass guitar in a rock band in 1970 before returning to keyboards in 1974 to play in a jazz-rock group. In 1977 he founded the free jazz group Freemprovisations, which featured Peter Schärli among its members; he formed Impetus in 1979. He studied at the University of Fribourg (late 1970s), Swiss Jazz School in Berne (1975-79), the conservatory in Lucerne (1979-84), and the Academy of Church and School Music (1985-87). His brother Peter Wolf was a singer, played saxophone, flute, oboe and founded in 1977 in Weggis, Lucerne, possibly the most successful progressive rock bands to come from Switzerland Flame Dream (1977-1986).
From 1980 to 1984 Brennan played in the Mohrenkopf Afro-jazz band, and worked in Impetus and Triumbajo with Ushma Agnes Baumeler and Barni Palm in 1980-82. Between 1982 and 1989 he worked extensively with Urs Leimgruber, and in the 1980s also worked with Corin Curschellas and Christy Doran. In 1988, he worked in New York City for six months, then founded the drumless quartet Pago Libre the following year. Early in the 1990s he worked with Lindsay Cooper, Daniele Patumi and Tscho Theissing in several ensembles and established the SinFONietta ensemble in 1991. In 1993, he worked with American drummer Alex Cline in the quintet Shooting Stars & Traffic Lights. In 1994, Russian hornist Arkady Shilkloper joined Pago Libre, resulting in a string of albums, from "Pago Libre" (1996, re-released 2002) to "Stepping Out" (2006), "platzDADA!" (2008) and "Fake Folk" (2009). In 1997 he lived in London and worked with Julie Tippetts, Evan Parker and Chris Cutler in a sextet called HeXtet, which set poems by Seamus Heaney, Edgar Allan Poe, Theo Dorgan e.a. to music. In 1999 he toured in Finland with Ivo Perelman, and worked with Gianluigi Trovesi, Gianni Coscia and Daniele Patumi in the quartet "Euradici".
As improvisor, he worked with Chicago bass-clarinetist Gene Coleman in a series of MOMENTUM albums, and with Christy Doran and Patrice Héral in the group Triangulation, where he developed his personal style of "comprovisation", a term he coined in 1989. He also released seven solo piano albums so far, from the first "The Beauty of Fractals" (1989) to "Pictures in a Gallery" (2006) and the award-winning "The Speed of Dark" (2009). Following his album "The Well-Prepared Clavier" (1998), he has developed an extensive array of prepared piano techniques, creating a whole universe of non-electronic sounds, from "arcopiano" to "pizzicatopiano", "tamburopiano" to "sordinopiano". In 2010, he created the sound installation "Inner & Outer Spaces" together with video artist Susanne Hofer for the Lucerne Art Museum, performing live with Gerry Hemingway and Thomas K.J.Mejer. Together with yodel singer Franziska Wigger and brass player Hanspeter Wigger he works in the trio Melos Montis, with pianist Esther Flückiger in the piano duo TwinKeys. In 2010, he launched the ensemble SONIC ROOTS with Andreas Gabriel, Christy Doran, Marcel Oetiker, Heiri Kaenzig and Marc Halbheer. In addition to his work in jazz, Brennan composes for film ("Meditative Moments", 2011, on YouTube) and the art-music world as well, particularly chamber music and songs and pieces for theatre, among them the two operas "Güdelmäntig" (2004) and "Night.Shift" (2007, based on W.H.Auden's poem "The Age of Anxiety". A revamped Pago Libre featuring Arkady Shilkloper, Tscho Theissing and new bassist Tom Götze performed 2012 in Austria and Switzerland. The first volume of his SONIC ROOTS series of books (for piano, inspired by Celtic Country Dances) was premiered at the Frankfurt Book Fair 2010, published by Pan-Verlag, the second and third (for violin) were published in 2011, the fourth (for clarinet) in 2013, the fifth (for alto saxophone) in 2014. In 2012, he started a collaboration with overtone singer Christian Zehnder and Arkady Shilkloper. The same year he recorded "Pilgrims", a new trio album with percussionist Tony Majdalani and guitarist Marco Jencarelli. The Percussion Art Ensemble Berne premiered his new composition "Oscillating Orbits" in 2013, scored for marimba, vibraphone, timpani and various percussion instruments, featuring violinist Misa Stefanovic. In 2015, the NOB Neues Orchester Basel commissioned "Traumpfade", a new piece for orchestra and overtone soloist Christian Zehnder. For the 30-year anniversary of the Zurich James Joyce Foundation he wrote "Winds of May", a new song for soprano and piano, based on Joyce's Chamber Music IX. He also wrote a new hymn for his Swiss hometown Weggis, called "s'Wäggiser Lied" (Text: Markus Wolfisberg & Josef Doppmann). For the "Alpentöne" Festival 2017, he composed "got hard", an alpine jazz suite for wind orchestra, Pago Libre & Friends (released on Leo Records 2018). In August 2018, the Tonhalle Orchestra Zurich (conducted by Kevin Griffiths) first performed his new and extended work "Traumpfade", with soloists Arkady Shilkloper (alphorn) and Christian Zehnder (overtone voice, global yodeling), at the Festival "Stubete am See".